This sounds very typical of carbon buildup in the engine/exhaust. The carbon build up is a result of the generator not having enough load for long periods. Using the generator to just charge the batteries is not good for it. It needs to be run with a 70% load, so turn on other stuff like the AC and do some laundry. If not addressed this will cause issues with pistons, rings, injector and valves and can be very expensive to fix.
Remove the exhaust elbow and look inside. This will serve as your indication that the engine has excessive carbon build up, or not. Black soot is okay on the edges. But, if it appears to be caked on the sides and is impeding the flow of fumes, this is a problem. Clean or replace the elbow if needed. Hopefully this will solve your issue because this is where exhaust fumes are cooled. However, you should also look inside the exhaust manifold of the engine leading to the elbow. If you continue to find excessive carbon in this area you may have a bigger problem.
Hope this helps
With best regards,
Sailing Vessel - Cream Puff - SM2K - #275
Currently cruising - Vista Mar, Panama