How about putting mattresses on the floor for your baby to crawl on? Can put two single size mattresses side by side, then cover using a king size bedsheet, and voila, you have an extra large 'playmat'. When not in use, just roll up or just place against the wall. I think two single size mattresses will cost less than a playmat. Not only that, the single size mattresses can be used by adults too, not just the baby.
For me, no matter where baby is playing (on the bed or on the floor, etc), adult supervision is still vital.
From what you mention, seems like your baby feels secure with someone around or someone carrying her. Also, maybe it's just the rooms give her different environment - can check whether any difference in airflow, room temperature, brightness, or maybe even noise level?