Relative to the engine even the largest generators are only 1/10 the horsepower or less so for generator cooling only it will not require much air flow.
I can’t figure a downside to sharing the radiator, especially since the engine cooling system is already shared, although it is to use the heat, not get rid of it. Today’s engines are running 200 degrees or more however and none of my generators exceeded 180 degrees so that might pose a problem.
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