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.