Performing the RTR around pets should not be a problem. But you should not clean your Aura/Soul around pets or kids, as they can absorb the negative energy you clean off yourself.
You can go to the bathroom and close the door to clean your Aura after the RTR and send the negative energy away when you're done. It would be good to also perform Aura cleaning on your cat (explained on the bottom of the page here:
Returning Curses ) and also put an Aura of protection on her/him.