Put a piece of cardboard in the bilge and see if it has been wet when you get back.
A Hydrometer will test for a dead cell (which will run down a battery bank). But since you just replaced the battery, it is less likely. There are infant mortality batteries occasionally so new does not always mean good.
A simple draw test is to loosen one battery connector and see if it sparks when you close the gap between the post and the clamp. I wouldn't recommend this on a system with delicate electronics though.
I suspect you might have a slow leak and your bilge pump is powering up often enough to hide the leak and the dead battery bank is a symptom, not a cause.
If you can't easily see down under the engine, use a laptop and a cheap endoscope to stick a camera where you can't see / feel as a remote set of eyes. most are water rated so it won't get hurt if you submerge it.
I would run my hands under and around all the raw water lines and where they connect to the engine and heat exchanger plus the shaft log. You might have a slow drip under the engine and not know it.