We originally had carpet, which the installers removed. I believe there is sub flooring underneath.

Thanks. I'm looking for solutions I can put on top of the carpet - doesn't look like anyone's got one besides the yoga mat.

Just put down a sheet of thin polyethylene between the carpet and rubber tiles/yoga mats to serve as a barrier for any sweat that you might not get to wipe up. Just tape it down along the perimeter of the room. They even make some plastic sheeting that is adhesive backed that will stick to carpet (people often put this down on the high traffic areas when selling a house). Put it wall to wall under your new pain cave flooring and your carpet should be fine.