I think there is some terminology confusion going on here.
The cooled dry exhaust that you describe is more commonly called a wet exhaust manifold and is very common on marine engines. In fact, they are almost universal.
The heat rejection spec for your engine should include everything in the engine coolant loop as supplied by the mfg. This would typically include the main engine water jacket, but also include the wet exhaust manifold. Now I say "should", but its always possible that the spec you have is for the non-marinized engine which would normally not have a wet exhaust manifold. Perhaps there are other clues in the engine specs about whether they are talking about the marine version of the engine? For example, if they talk about the sea water pump, then the spec is surely for the full marine engine and would include coolant heat rejection from the wet manifold.
In contrast, the gear cooler would typically NOT be part of the engine heat rejection spec. However I expect it is minuscule compare to the engine, so not a concern from a heat rejection stand point.
As a reference point, a modern Deere 160HP engine rejects about 350kBTU to the coolant circuit. So something in the 250KBTU range for your engine seems reasonable.
Another point of miscommunication I think has to do with the bypass circuit. Cooling systems have an internal bypass path to allow coolant to continue to circulate while the engine is warming up. You shouldn't screw around with that in any way, and as has been pointed out, using the wrong type of thermostat can defeat the bypass causing all sorts of problems.
Separate from the bypass is the aux heating loop commonly used for heating domestic water and heating the cabin. It has nothing to do with the bypass circuit. The aux heating loop is the thing that is optional and can b valved off or plugged if desired. Manufacturers will also commonly like flow through the aux loop using fittings with fixed opening sizes, and limiting the size of allowed hose, all to prevent over cooling of the engine. I cant tell which of these you have modified, but neither may be a good idea.
BTU specs are one thing, and as you have surmised, unimpeded coolant flow is another thing entirely, and yours indeed seems suspect. A 2" engine hose size, going to 1.5" for the manifold is not great, but I see nothing you can do about that. The subsequent drop to 1.25" for the gear cooler, hose runs, and keel cooler seem very problematic.
But before making any changes, I totally agree with DMarchand and Ski about diagnosing the problem before you plan the remedy. Unless you are getting hot coolant out of the keel cooler, you should assume the cooler is fine.
Another good test would be to measure the coolant pressure somewhere between the pump outlet and the entrance to the gear cooler. You mentioned calculations in the 25 to 35 psi range which is nutty. Specs I've seen call for no more than something like 5 psi back pressure through the external loop. I think your suspicion is correct that you just aren't getting sufficient flow. But temp measurement along the path will confirm one way or another.
I suspect that if you bypassed the gear cooler and ran 1-1/2" or even 2" hose to the cooler that it would work fine, even with the step down to 1-1/4" through the cooler itself.