Most of your lights use 12V DC from the Power Converter and Battery. If they flicker when the heater uses 120V AC, it's signaling several possible problems. I'd check first to ensure that the battery power is getting to your fuse box. The easiest way to do this is to go into the main circuit breaker panel, look and turn off the breaker for the Power Converter. On Landmarks, that'll be a 20 amp breaker. If your lights go out when the breaker is off, you need to reset a 12V DC mini-circuit breaker near the batteries. There's a row of them covered by red rubber boots. One has a teeny tiny reset button on one end.
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If the lights flicker or go out altogether, you have a problem with the Power Converter.
If lights stay on, you may have a problem with the 120V AC wiring/components. You could have a loose connection inside the automatic transfer switch that shifts power between generator and shore power. There also may be a problem with the Power Converter.
Turn the breaker for the Power Converter back on. Then use a volt meter to monitor your 12V DC voltage while the heater is running. I'd suggest removing the cover from the fuse box and measuring where the power converter connects to the fuse box. Use the ground bus in the fuse box as the ground connection. Before testing, turn the 2 battery cutoff switches to the OFF position. That way you'll measure Power Converter output only. If the voltage fluctuates when the lights dim, you either don't have sufficient voltage to keep the the Power Converter running, or the Power Converter is starting to fail.
Let us know what you find.