I have done it, in fact I have traveled three nghts in a coach changng in Chicago, coming from the southeast, but many years ago.
So, I lived to tell the tale. I am sure the people who deboarded before I did were not just trying to get away from me!!
Major point I would make is that there is nothing inherently dirty about train travel. Taking a good shower or tub bath before going to the station will go a long way toward keeping you at least "presentable".
If by "bird bath" you mean stripping down and cleaning yourself with soap and rag, not sure. One used to do that in sleeping cars, before they had showers on a large scale. My mother called those "sponge baths"--I guess it is the same thing.
Somehow I am not sure how workable that is on a coach rest room today. Hopefully others with more current experience will help.
Still, you really should not aquire too much additional dirt! Not like the old open window/steam engine days!!