If they sleep in a hostel where there are several people in the room, then seemingly this is not just minimizing the prohibition — there is really no prohibition here at all, no?

The prohibition is not seclusion but entering a situation of forbidden thoughts. Sleeping with a woman in the same room is not just lodging, plain and simple. Still, I hadn’t noticed the possibility of sleeping in the same room with other people. If we’re talking about sleeping with other people in the same room, then in my opinion there really is no problem with that.

The Shulchan Arukh wrote in section 306, paragraph 14, that it is permitted to desecrate the Sabbath in order to save a Jewish girl from forced conversion and religious persecution. And the Mishnah Berurah wrote: “For when she completely converts, she will desecrate the Sabbath and worship idolatry all her days; and if he desecrates the Sabbath one time, this is considered a minor prohibition compared to that.” So we see an explicit halakhic ruling that one may violate a small prohibition in order to prevent a greater prohibition. What does the Rabbi say? That one violates a minor prohibition only in order to prevent conversion, which is more severe? Or perhaps one should distinguish, because here he desecrates the Sabbath for her, and she is not the one desecrating it. And accordingly this falls under the category of ‘he cannot restrain his inclination,’ since he has no control over her. What does our Rabbi say?

Tosafot and Rashba disagreed about this, and Rabbi Yisraeli already discussed it at length in Techumin (issue 1 or 2).
I don’t think the intention here is the distinction between a major prohibition and a minor one, about which the commentators wrote at length around the case of sticking bread into the oven at the beginning of tractate Shabbat. Here it is something like saving life — spiritually.
There is no difference in Jewish law between desecrating for someone else and his desecrating himself. Check carefully the disputes among the medieval authorities (Rishonim) regarding feeding a sick person carrion or properly slaughtered meat on the Sabbath, where none of them (except for the initial suggestion of the Tashbetz, which he rejects) even raised the possibility of distinguishing by saying that with slaughtering I desecrate the Sabbath for him, whereas with carrion he violates the prohibition himself. So we see that there is no difference. And similarly we find in the responsa of Igrot Moshe and Achiezer that they rejected the view of Mishnat Chachamim regarding performing a dangerous surgery in a case of possible danger, where Mishnat Chachamim said to give it to a non-Jewish doctor, and they rejected his words because there is no difference between him and others.