My wife and I are members of a hot yoga studio and take classes several times a week. We have for years. All the teachers are Americans and the classes are purely secular. But it is a great workout and it has many health and psychological benefits. The heat in the studio allows you to get deeper into the poses without injury.
Saying Catholics can’t practice yoga is like saying they can’t compete in the Olympics, because The Games were dedicated to the Greek god Zeus. The medals given out since 1928 have the imprint of Nike, goddess of victory. Also Catholics could not practice martial arts because of its roots in Hinduism, Buddhism, Shinto, etc. Not to mention that a lot of the sports we play were invented and/or popularized by protestants!
To the OP’s point about Downward Dog: Your heels touching is not the key to the pose. There are about ten keys to that pose that have to do with alignment and where the tension and relaxation is. The heels should be pressing toward the mat, but it’s OK if they never get there. You are getting the benefits of the pose anyway.
And don’t fret about the “contortionists” in yoga. The great thing about the practice is the less flexible, the less strong and the less balanced you are, the more benefit you are getting out of it. The contortionists are sort of “cheating” and should be doing more complicated poses that challenge them as much as the basic poses challenge you. It is not a contest.