Conservative means wanting nothing to change. Whatever society is, it should stay the same. What that means in practice depends on where you come from, but it is often, and in this case specifically, related to the position of the church.

Make it part of the religion

Consider the historic concept of the nun or monk under Catholicism and invert it. They're no longer wedded to god, they're now wedded to society and sex is a holy act. One makes a donation to the church and the church in return grants one time with a nun or monk, as appropriate, to perform holy rites commensurate to the amount of money you've donated.

This isn't extra-marital sex, it's a form of worship.

It's not a guild, it's a church, and religion is socially powerful.

And those who have the favour of the church are also those who clearly have the favour of god, i.e. the wealthy and powerful of society.

Temporary marriages. They already exist in this world. It's a private contract for a pre-determined amount of time. Agree the terms, get married, have sex, marriage ends.

https://en.wikipedia.org/wiki/Nikah_mut%27ah is the name given to this practice in the largest sect of Shia Islam.

A Sunni Islam option also exists; https://en.wikipedia.org/wiki/Misyar_marriage . It has no fixed time period and involves the man paying the woman.

There is also https://en.wikipedia.org/wiki/Nikah_%27urfi , which apparently specifically has a reputation of being "a way for people to have sex with each other within what is perceived to be a licit framework." That sounds exactly like what you need.

Be practical: if you are strong in your statement that only intra-marriage intercourse is legitimate, you have a huge problem.

You have a large mass of single males and females, mostly in their hormone storm age, with no access to "relieving practices". Any respectable member of the society would dread the idea of their son/daughter being raped, hence losing their honor and value as exchange matter, before marriage.

So, better have brothels where steam can be let off. Since they contribute to the overall well being of the community, and also pay taxes out of their profession, members of the guild are respected, and being a competitive market it's to be expected that competition will happen not only on physical appearance but also on side entertainment, highly valuing the ability of having philosophical conversations, for example, for which a dedicated training is required.

At the very end is nothing different than what was happening with prostitution in real history: see the Greek Hetaira.

Hetairai were thought to have had only a few men as clients at any one time, to have had long-term relationships with them, and to have provided companionship and intellectual stimulation as well as sex.

Sex workers are not (regular) humans

That's right. If your society treats guild members as someone detached from humanity, sex with them no longer is "real" sex. Making them sterile would help a lot. Making them a kind of monastic order that answers to "pope" would also help a great deal.

In human history we used to have societies with castes treated like they were something totally different, so this wouldn't be that far from what your readers would probably already know. Think, for example, third gender:

Thus, in ancient India, as in present-day India, the society made a distinction between a third gender having sex with a man, and a man having sex with a man.

As you can see, societies can make such distinctions already, what I propose is quite similar.

Source: https://en.m.wikipedia.org/wiki/Third_gender

What's formally acceptable and what's informally common practise are not the same thing.

You are basically describing the state of prostitution in Europe in the 17th-19th centuries. Strict Christianity was the foundation of moral society - but if you were female then your career options were basically domestic service (where you'd be expected to be sexually receptive to your employer), marriage (where you'd be expected to be sexually receptive to your husband), or whore. And there were a lot of whores, and men thought no more of visiting them than they did of visiting an alehouse.

You might think that strict Christianity would lead to laws limiting this - but of course the people actually making the laws were the same ones visiting the prostitutes. Even so, they had to be seen to be doing something, so some laws did exist. Except that until the foundation of modern policing by Robert Peel, enforcement of those laws was purely by "thief takers" paid by magistrates who themselves were rich, and who would not cross another gentleman (or their friends). Hence the word "privilege" for "private law" - not only was enforcement of crimes against people subject to the depth of your wallet or influence, but so was the choice of which crimes were pursued.

Successful prostitutes could move up the ladder to be "courtesans". If they could get the favour of someone prepared to pay for their sole services, then the mistress of a rich or influential man could even have substantial standing herself, and this would not be hidden from the world. Think of Nell Gwynn, for example.

Ironically, despite this level of public knowledge, the worst that anyone could do was to record this formally in writing. Hogarth, Fielding and Dickens scandalised English society by touching on this kind of thing in their work - even though everyone knew what was actually going on.

Give "Intimacy" another meaning

"The only legitimate form of sexual intimacy is between husband and wife."

Then let your religion define "Intimacy" as "Sexual Intercourse For The Sole Purpose Of Procreating". From there, any sexual act that does not involve procreating is considered as an... inconsequential physical entertainment.

Like L.Dutch said in his answer, brothels are places were the steam can be let off, and everything happening between these walls are considered as benign because nobody gets pregnant.

Of course, if any prostitute gets pregnant, the Church intervenes to find and bring the guilty parties to justice...

Similar to "Companions" in the Firefly series, the prostitutes are highly skilled members of a professional guild and have a great deal of wealth and autonomy.

A Companion is a skilled, well-educated and well-respected member of a guild of professional courtesans/entertainers, somewhat similar to oiran. While they do frequently engage in a form of state-sanctioned prostitution, they are nonetheless treated with a great deal of respect and deference by nearly everyone,[1] though non-guild prostitutes[2] are generally treated in the reverse, even if they are former Guild members.

Consider also religiously permitted temporary marriages which are accepted by some in the Islamic faith.

According to Twelver Shia jurisprudence, preconditions for mutah are: The bride must not be married, she must be Muslim or belong to Ahl al-Kitab (People of the Book), she should be chaste, not addicted to fornication and she should not be a young virgin (if her father is absent and cannot give consent).[9][need quotation to verify] At the end of the contract, the marriage ends and the wife must undergo iddah, a period of abstinence from marriage (and thus, sexual intercourse). The iddah is intended to give paternal certainty to any child/ren should the wife become pregnant during the temporary marriage contract.[6][1]

Giving the religion significant power over the relationships will probably help in making them more plausibly acceptable. A self-regulated professional group under religious supervision might work well, similar to doctors, engineers and lawyers under secular governments. Unauthorized persons who dabble in the profession are subject to severe penalties under the law.

I'm going to use securiger's comment for this: https://en.wikipedia.org/wiki/Sacred_prostitution

Even in the Christian religion you can find forms of prostitution down to Jesus supposedly protecting one and allegedly even loving one, and the Koran also basically rewards people with prostitutes with the whole "72 virgins" thing. with sacred prostitution being a thing you could argue for a society who attempt to remain pure by having people release their sexual urges by sanctioned and possibly "blessed" prostitutes. This way they can avoid the "only sex after marriage" problem.

Use prostitution as an outlet within a legal marriage, one that enforces conservatism in other ways

It's natural for men's eyes to wander, but god insists we keep a standard family unit and don't divorce. So men who aren't sexually fulfilled are allowed to go to the prostitute so as to not break their sacred vows. The social stigma of prostitution would go to the wife, not the husband or sex worker, for failing to meet their husband's needs, thus keeping gender inequality firmly in place.

There are a number of ancient religions where priestess and courtesan were the same, typically for the goddess of love or similar.

Add a bit of business sense to this concept and remove the necessity of the sex act happening within the temple (just within the priestess) and you have essentially an escort service that donates its income to charity ^H^H^H the temple.

That concept is absolutely compatible with a religious and conservative society. Conservative does not necessarily mean prude, that is just a combination typical of modern America.

Make premartial sex immoral. Then teach that prostitution is a "necessary evil" to prevent lusty men from taking out their lust on innocent girls.

This was the case in the Middle Ages and 19th century France and Russia.

Medieval civilians accepted without question the fact of prostitution, it was necessary part of medieval life. Prostitutes subverted the sexual tendencies of male youth, just by existing. With the establishment of prostitution men were less likely to collectively rape honest women of marriageable and re-marriageable age. This is most clearly demonstrated in St. Augustine's claim that "the removal of the institution would bring lust into all aspects of the world." Meaning that without prostitutes to subvert male tendencies, men would go after innocent women instead, thus the prostitutes were actually doing society a favor.

St. Augstine was a Bishop and theologian.

France, instead of trying to outlaw prostitution began to view prostitution as an evil necessary for society to function. France chose to regulate prostitution, introducing a Morals Brigade onto the streets of Paris. A similar situation did in fact exist in the Russian Empire; prostitutes operating out of government-sanctioned brothels were given yellow internal passports signifying their status and were subjected to weekly physical exams.

Prostitution and sex crimes aren't uncommon amongst religious leaders today either.

India's devadasi girls are forced by their poor families to dedicate themselves to the Hindu goddess Renuka. The BBC wrote in 2007 that devadasis are "sanctified prostitutes".[88]

Separate true religion from socially acceptable religion

In your hypothetical society, perhaps the conservative social leaders and trendsetters have all the appearances of religious conformity but are actually sin-deniers, believing that there's no problem really with prostitution, especially expensive prostitution, the way they do it.

In this conservative society, the trend-setters and leaders would live every day as if they were religious, but there would be some people (maybe you could name them "Grablers", "Pleamlings", and "Zarbs") who could see that there was a disconnect between what they said and what they did. Maybe the Grablers would scoff at religion while the Pleamlings scoff at the leaders; and wouldn't it be dramatic if the Zarbs take a stand for the truth at their own personal peril?

well, it have been in europe, at least in france: mastubation was so badly see that the church accepted that men go paid a prostitute each time they where feelink like jaking off. it wasn't considered like cheating, because it where their job, and religion couldn't forbide both, since men needed way to enjoy themself.

so, ban masturbation and because they will be the lone socialy acceptable way to enjoy yourself they will quickly gain a lot of influence.