So, if I read this correctly, if house batts are good, gen starts. If main engine is running gen starts, but if neither is good then gen is a no go?
If so, the gen battery (or connections) must be bad even though it has voltage, it may not be anything but surface voltage. I assume that you have a way to tie into the house batts as well as the starting batt for gen set? It would appear that is the case as it will start with the above listed requirements. You must have a switch that is allowing the house to power generator starting circuit. When main engine is running, it is flowing volts to house, so that is probably why gen will start.
On my Legendary, I can tie into house circuit for starting as well as dedicated generator batt.