You can use carpet (and many cats did in fact use it for headliner
material). Indeed, my boat was originally fit with both the glue on vinyl (in the heads/galley/main saloon) and carpeting in the staterooms. I still have it as a liner in some of the closets.
The glue on the carpet can fail, but more significantly, I found that the carpet did tend to develop mould when the boat was closed up for a lengthy period. Still, better than the foam backed vinyl.