I trust you have three prong plugs and are sure you haven't cross a neutra.
Larry, why not just wire the inverter in correctly and be done with it?
I'm not sure what it might be a function of (something as stupid as an electric tooth brush's charger) or a laptop charger that is "loading up" all of a sudden.
You've checked the neutral to make sure you don't have something thrown into the ground?
You really need to have a manual transfer switch in there. They make them for a shore-off-generator scenario. You could simply use something like that to power your convenience outlets from one or the other.