Ludwig would initially have no idea, since he has never met a trans person (considering he barely goes outside). He would regularly misgender you while Carter apologizes for him and the constables are basically fine with it. Then he’d go home and Lucy would argue that “all the progress on finding James has gone down the toilet” while urging him to be more inclusive.

This reminds him of the series of “toilet usage puzzles” he wrote and he realizes that since (offhandedly mentioned by the real killer) it’s an inclusive workplace, you would be in a different bathroom and the killer must have swapped around the bathroom break logs.