"Rule" is a maker of small 12V pumps. I have one that is about 3.5" in diameter and might fit through a port (There's always the option of installing a larger port, if you think that this is a recurring issue). If you don't have a store nearby where you can look at examples, try Fisheries Supply.
I got something like this model:
https://www.fisheriessupply.com/rule-360-gph-bilge-pump-with-float-switch-24-35a.
Now, you may not need / want the float switch - the way I understand your application you would normally check and activate the pump manually anyway. I've used this combination as a sump-pump in an external stairwell for years - an application where battery operation is a plus as flooding occurs more often during power outages. . .
Either find the same model w/o the switch or just don't use it.
All that aside: if there's water in your bilge that you can't account for, I would haul the boat out anyway to make sure I understood where it came from. . .