Will the engine restart after it stops and before recharging the battery? Or does the battery need to charge 1st?
The engine disconnect relays normally require /both. power and the pressure switch being closed to /enable/ the engine. If they lose power, the really is normally open and the engine stops. They are normally powered off of whatever powers the ignition circuit, usually the start battery, not an unrelated battery such as house or genset.
Electric fuel lift pumps are also a possibility, if someone installed them. I think the C9 has mechanical one's natively. But, people do like to install electric ones when they fail. If there are electric fuel pumps, they could be a problem if they are wired to the wrong battery.
Lastly, there is, almost literally, everything else. By (fallible) memory, the C9 is a modern engine with a billion electrical sensors, solenoids, and an ECM. It isn't like an old school Perkins, or Lehman, or Detroit. It is more like a gasoline engine -- no power, no run. It is hard to imagine how it would end up miswired, but anything is possible. (Don't hold me to this, it is just what I remember).
I'd start by disconnecting the genset battery (red wire is fine) and seeing if the engines start and run. If they don't, then I'd see what has lost power, e.g. the fireboy or the lift pumps, or, ... And, then go from there to get any engine-related systems moved over to the right battery.
If the engines do start and run without the genset battery, then I've got a million more questions, some of which may not have apparent answers.