I'm thinking your question would be, "Does a dead battery effect the output of the converter?"
Typically, a converter will output the most power when a battery is discharged. In that event, with shore power connected, the converter should be able to supply 12 V to run lights, pumps, fans, etc.
You are really only guessing at all this until you have a meter in hand to measure actual power. If you can't power the lights with just the battery connected, the battery is dead or the circuit breaker between battery and distribution panel is open.
My suggestion would be to leave the converter off and use the charger 24/7 charging the battery, while also providing 12 V power for lights, fans, the A/C, etc.