[Richard]

So you say 13.6 at the starter terminal on the gennie. If that is so
Don't chase battery issues.

Focus on the starting circuit that will pull the solenoid in for the gennie starter. Do you have 12v at any of the generator start switches? Will it start from the panel on the generator?

If you short the starter solenoid to the starter terminal *otherwise known as hot wiring" will the gennie turn over? It probably won't run this way because the fuel shutoff solenoid won't activate, but it would at least eliminate the starter as the issue.