You didn't mention it, but the other thing to check is the house battery condition. Even though it's not looking at the voltage until you release the start button, a battery that's on the way out will cause all sorts of strange things.
Get some one to help you and put a digital voltage meter across the house battery, measure the voltage, then with the meter still connected, have the other person crank, start and (hopefully) run the generator. If the voltage drops much, repeat with a good battery connected by jumper cables. See if there's a difference.