We tried to make a dry bilge but it can't work for the same mast reason (keel stepped mast allowing rainwater to enter).
I would think that shower water has the potential for a nasty bilge. Why not a sump pump or just let it slosh in the pan until you pump it overboard with a manual switch & pump. I always liked the idea of a manual switch and pump as you can turn it off at the main panel and tell guests that their shower is over when water is over their toes. It was about 2 gallons for our shower pan and a kind of mandatory water conversation method for guests that tend to use a lot of water for showers.
I guess the flip side is that it's simple and soapy fresh water in the bilge might be better than salt water and oil.