Actually, getting them used to the carrier is an excellent thing!
I leave the carrier open in the dining room at all times. Dancer and Velvet hop in and out of it, too. Dancer loves to be in there. I have a pillow in there and he digs around in it. So funny! When I need to take them to the vet, they are not afraid of being in the carrier. I always leave that pillow in there so they know it’s ok to be in there and there is nothing bad that will happen to them. The carrier gets connected with positive things here. They either get to be outside on the lawn when I carry them out there in the carrier or they get to be with very loving people at the House Rabbit Society.
As far as the AC is concerned, yes, the bedroom will work just fine. It’s great that your husband can go check on your bunny (and also to make sure the air conditioner is still working properly throughout the day). What a lucky rabbit girl!