I had a Hallucination on this. Going along with the theory that the battery is always the problem , could it be that it is not getting a proper charge while running, to bring it up to a proper strength to spin the engine? The generator regulator system may not be working correctly and only providing enough juice to run with some help from the battery. After just reading what Bartamos said, I think the load test is in order as well. Also check to see if there is a draw on the battery that could be taking some power away. I believe there is a simple test for your generator, regulators are still a mystery to me but if you get a good test from the generator, you will need to look at the regulator and confirm it can charge the battery. If your lights dim while idling that might be an indicator something is off in the regulator. Again, an hallucination.
