I thought about that, but the battery is only 2-years old. I also checked to see if the battery was sinking current into it while the generator was running so I put an ammeter in series with the battery and there was only aboy 10 ma (1/100 of an amp) flowing to the battery.
I think all these sorts of air-cooled engines use what is called a permanent magnet alternator to supply current (mine generates 28VAC). I had to troubleshoot the one on my twin-cylinder lawn mower engine last year (turned out to be the rectifier / regulator). These devices incorporate a magnet built in to the flywheel, and the pickup / coil mounted to the side. Since my mower, once running, will operate perfectly well without the battery even being in it, I must assume that no battery current is required for motor operation. While the operation of the Generac OHVI engine may be somewhat different in detail from my engine (A Kohler Magnum), I suspect that as described it's about the same.
I would be looking to the battery or its connections....possibly a weak cell?