As a student, how can I negotiate comfortable and appropriate sleeping arrangements with a stranger for a conference? I'm an undergraduate student that was offered a role researching for a grant. I was able to get authorship status on a paper and will be attending a conference for it later this year. The professor I am working for connected me with another student at a different university in order to share a hotel room. We will both be staying for the entire week and are responsible for booking our own reservations. Unfortunately, the centre the conference is located at is booked, with only a single-bed room available that we would be required to vacate on Wednesday then move back in.

This student was fine with booking the room and asked me if I was okay with it, but after I realized that it is a very small, dorm-like room with a small bed and no futon I counter-offered a reasonably-priced room at another hotel a half-mile away with two beds that we would not need to move out of midweek. I thought this would be well-received, however the student pushed back saying that the other room had a better price (of course, as it was intended for one person or a couple) and we would have better involvement with the conference (doubtful, as long as we show up on time).

Besides already being an incredibly poor sleeper, I'm otherwise just uncomfortable sharing a small bed with another student I don't know, and the room is small enough that I can't confidently say a hotel cot would fit. However, I also don't feel comfortable making this a bigger conflict than necessary, and I don't want to upset my professor by costing our grant more money than necessary by booking my own room at twice the cost. On the other hand, I do feel I have the right to request sleeping accommodations that I am comfortable with. How can I handle this appropriately without offending anyone?
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.