Could you temporarily put the mattress on your bed for a few days? (I've seen people "store" them in this manner before) However you do it, you're on the right track to figuring out a way to get it done. Maybe you could set up in another room for a couple of days? How about the kitchen table? Maybe even a porch or patio outside? A local library or friends house? You keep considering your possibilities. You obviously want to finish this quilt for your friends, so the drive and determination are there; all of the rest are minor inconveniences to overcome.
Did you find those other patches?