Okay, my old lessor owned several properties and a tenant at one of them was picking up dog poo and placing it on the lawn and covering it with dirt. She had serious mental health issues. There's a world of difference between that and a facetious claim that someone claimed they were a cat and wanted a cat litter tray.
The right-wing need to go by the principle of "Your right to swing your arm ends at my nose," and say, "If someone is trans, it is their right and their business to express themselves as they please, and I have no right to contradict them. I can have an opinion, but I cannot deny them their right."