That is a fair idea.
The only challenge is that hair and soap scum can accumulate in the bilge.
"Organic material (hair, soap residue, skin cells) decomposes in standing water, producing the classic “boat smell,” mold, mildew, and bacterial growth. This is especially common when shower water drains directly into the bilge rather than a dedicated sump." (Internet search for effects of accumulation in a bilge)
Hair does not dissolve. It tends to clog pumps and screens. Soap scum acts like glue and can plug your bilge pump; the combination often leads to pump failures.
Best practice is to use a separate sump with a dedicated pump that you can clean out from time to time.