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.
I train many people with pets. It becomes a matter of if the pets are truly in the way or if they stop by to see what is happening once in a while. I have never had a problem with working in clients’ homes, but as a trainer you do let them know ways to stay healthy, which includes living in a stress free, well maintained home. People do love their pets, so remember to be careful with your word choice. If possible you could always take your client outdoors (I utilize hills or steps of people’s houses/blocks). And, with dogs that is a way to incorporate extra cardio in someone’s routine; encourage them to take more walks/hikes with their dogs on off days.
Maybe suggest you train them in their backyard or ask them nicely to have their pets elsewhere while you train. Tell them you can’t give them your full professional service when there are distractions, so for them to get what THEY paid for it should be easy for them to make the proper adjustments.