As a student, how can I negotiate comfortable and appropriate sleeping arrangements with a stranger for a conference?
You don't mention what country you're in and perhaps it might matter, that what's acceptable in one country might not be in another. Here in the US, it's not uncommon to be asked to share a room at a conference or in other professional situations with another attendee of the same sex. But it would never be considered appropriate to ask anyone to share a bed. Never. Period. It would only happen if a supervisor, e.g., your department chair was unaware it was happening.
So, if you're here in the US, I would be firm. Send an email to your professor, pointing out that sharing a room is okay, but sharing a bed is not and insist on your solution, the room with two beds at the other hotel. I would not feel guilty even for one second about the extra cost. You're entitled to your own bed. If your professor insists you share a bed, I would report this to your department chair and request help.
The professor I am working for connected me with another student at a different university in order to share a hotel room.
I think this is the key point. The professor, who is your supervisor and is in charge of the grant funds, is sending you to the conference, and it is their responsibility to ensure you have safe, reasonable accommodations during the trip. Their proposed solution involving shared accommodations with the student they connected you with might have seemed reasonable when they suggested it, but it is becoming clear that it isn’t. Certainly asking that you share a hotel bed or other uncomfortably close quarters with a complete stranger is way outside the norm in almost all areas of academia (the only exceptions I can think of being a few disciplines like archaeology, paleontology and such where hardy researchers sometimes go on field studies in very spartan environments), and not something you should be expected to accept as a condition for attending a conference.
The conclusion is simple. You should email the professor, explain that you have not been able to find an acceptable arrangement to share accommodations with the other student, and ask him to help you find an alternative solution involving an acceptable level of comfort and privacy. Such a request, if phrased politely, would be completely reasonable and professional, and I’m sure any half-decent professor would not be fazed by it or have any trouble addressing the problem.
Good luck, and have fun at the conference.
You may just need to come to the conclusion that no negotiation is possible and make your own arrangements. No one should be offended by someone who wants to preserve their personal space and privacy.
I doubt that your professor, who is the only one, other than yourself who matters much here, would think less of you if you don't yield to a situation you find somewhere between uncomfortable and untenable.
There is the financial issue of course, but your privacy is likely worth the cost and you will have a much better experience at the conference if you are comfortable personally. While cost isn't a factor for me anymore I often tell family (cousins, kids, etc) I'd rather get a hotel than use their guest room. Privacy, comfort, personal space.
That said, you might take a bit of time at the conference to see if the other student is someone you might want to be friends and colleagues with generally, but that is a question for the future.
Due to a comment, perhaps I should be more clear and explicit. This is not something you should accept if you have any reservations. You are perfectly correct to reject it.