When the generator is running, the converter should be supplying 12V power to it (and the battery). I don't know if the generator requires 12V power while running for the control circuitry; probably depends on the generator model. It stands to reason that some, at least, might.
Do check that your battery is indeed charging whilst on shore or generator power--that is, that the converter and wiring between it and the battery are working properly. No battery charger will lead to the battery being run low if you aren't careful, and that will lead to battery damage if the battery gets discharged too deeply too frequently. (Also verify that it isn't putting out too high of a voltage and cooking the battery....)
If the generator is slow cranking when the battery is fine and fully charged, it could be an electrical or ground connection to the generator, a faulty starter on the generator, etc.
