If it is a safety issue, you need to address it. I hope the animals are pets (I draw the line between pest and pet), and here you can additionally claim a concern for THEIR safety because you do not want to drop a dumbbell on Fido.
Cleanliness – unfortunately – is often in the eyes of the beholder. Having a dog myself, my house may not hold up to meticulous scrutiny. I would just bring my own mat and ignore the rest.
My advice, though, is given from the vantage point of an animal lover. My dog is with me when I train my clients at my studio, and he is a very important component of my training.