What I would do is connect meter to battery terminals, with shore power connected and all circuit breakers "on", trip each circuit breaker individually until observed voltage drops from 13.5+ back to resting battery voltage (it should be less). This will tell you which circuit breaker is powering the converter. It may be that other circuits are also powered by the circuit breaker for the converter. That, you should also check.
The converter you have is kind of substandard for a matched pair of $800 AGM batteries. I would rather have a converter that would better fit the charging profile of the new AGM's. Assuming that it also doesn't put out a high charge rate, it will take a long time to recharge to an acceptable point. I'd suggest a new converter would be the best solution. You may also wish to upgrade the wire from converter to batteries, AGM can accept large charge current.