Mine always slept on one foot so perhaps your grey is getting a good night's sleep (and stays balanced) at least at night on a higher perch.
In mine's travel carrier (kitty carrier) I would roll up a small towel for a ground perch -- I taped the two sides of that towel perch on its ends (not the middle where parrot 'perches') to the bottom sides of the carrier with duct tape, making absolutely sure there was no tape lifting where a parrot foot might get caught. This won't solve the issue of her wanting to perch up higher, but at least, on the bottom of her cage, she would have something for her feet to 'grab' onto. Maybe something like this taped to a large flat square of cardboard that could be removed/exchanged for cleaning, towel-perch re-taping, etc..
Just my way of making a ground perch -- I never even thought of using a rope perch. But I did tape mine so was stationary -- if my make-shift towel perch had any movement at all, I think my grey would have been nervous. Yannoe parrots, one scare and they'd avoid it forever. All my parrots had this rolled up towel ground perch in their travel carriers when we moved 300 miles. Good for vet visits too. Dunno know it would work for you or not. Sorry.