Instead of 'on top of' where it could roll: I'm thinking, what if this rope perch could be suspended just 2-4 inches or so above cage bottom? Still not the height a parrot wants, but a fall would simply be more of a 'step off' thingie. Best of both worlds: low but a real perch.
My rolled up towel did sit on carrier bottom, but yours could be safely suspended at a very low height.
I know, I'm thinking and worrying over this too much. But poor little Corey.