Could be the converter or the battery
Here is how a battery can do it.. Shorted cell.. When the circuit breaker in the battery positive lead closes the battery draws way more current (due to being only an 8 or 10 volt battery instead of a 12) then it should dimming the lights.. Then the breaker overheats, opens up and the lights brighten. This repeats every period of time.
Or, it could also be the converter.. You do not know till you test everything
Simple test: Hook a digital volt meter to the battery terminals and observe
IF battery voltage less than 13 goes up when lights dim.. It's the battery
IF battery voltage goes DOWN when lights dim.. Converter.