After I posted, I realized that if she does any fiber work: knitting/weaving/felting, she likely uses the dryer to *full* the finished product. This is a final stage in all the fiber fabric crafts and it causes the wool to *bloom* (ie: get softer and have a nice final hand), plus it adds just a bit more shrinkage to any felting. Even on low heat, there would be lint. But you’re right about the angora and even the alpaca and llama: they just are attracted to all other fabric.
Cat hair is smooth (no little barbs,and it doesn’t shrink like wool) and so does not felt. It does matt, as any cat parent knows. So, if you bought some felting needles (fine ones), you could place your mat on a thick piece of dense foam (to absorb the needle strikes—those needles can be deadly) and just keep needling the accumulated cat hair into the mat. The thing is that needled felt isn’t as well fixed as wet felt, even when everything is wool. Your cat could just pull the hair off the mat by kneading or even rolling around on it. I have seen some cat hair needle felted onto sculptures. The synthetic felt available commercially and most industrial felt filters are needle felt, but that is done by machines made to tangle the fiber completely and then needle it to itself. I have read that needled human hair was/is(?) used for peanut butter production filters.
Best I can think of would be to 1) needle the shed hair into the mat and 2) then lay a thin fresh layer of any wool on top, needle in place, then wet with hot water (opens the fiber), roll up as you observed (provides compression) and then roll the place mat package around (provides pressure/agitation) until everything is adhered. You need to periodically unroll and check the process. I use mosquito netting and those rubber shelf liners for boats instead of a bamboo mat because you can see through the netting and only have to unroll the shelf liner a bit to check. Or ask your fiber friend to add the layers. She’ll know how.
/johnny