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I also suspect a the heat exchanger at the front of the engine to have a leak and thus when the engine begins to heat up and the pressure inside the cooling system begins to build the fresh water/coolant is forced out the sea water. This would go un-noticed except for the level of the coolant that you are seeing.
To test you will need to rig up a pressure gauge and valve to pressurise the cooling system while the engine is off and cold. You can by a pressure testing gauge for about 15 bucks at any tool town, and then make your own adapter for the cooling tank top. This is often done by drilling out a wooden bung (often on board for emergency plugging of a through hull) and putting the wooden plug into the cooler top. Then pump up the cooling system to 10psi and it should hold that pressure overnight.
One other thing that could cause this is the coolant top may be letting water escape because the pressure spring has failed thus without pressure the coolant will not hold in the tank. You can check this by exchanging coolant tops from the other engine. Like you say the coolant would them be in the bilge, but it's a simple test that does not cost anything.
Let me know how that goes.