I would suspect that you are actually running your battery down to about 30% capacity or lower during those 2-4 week periods. I don't know what 3-4 lights on the board means, or how accurate the lights are at measuring voltage. As soon as your converter comes on line, the lights are meaningless. The life of your battery also depends on what type of battery it is.
A starting battery is only good for 4-8 cycles down to below 50% capacity - then it's toast. A Sears Marine Diehard "deep cycle" is supposed to do 200 cycles to 50% capacity, which gives a reasonable lifetime on boat/trolling motor use. True deep cycle batteries (golf cart batteries are an example) will often do 500+ cycles to 50%.
The 50% is somewhat arbitrary. If 40% is your minimum, you will have fewer cycles than at 50%. 60% minimum charge gives more cycles than 50%. Deep cycle batteries are rated at 50% because of the trade-off in battery life vs usable capacity (the curve is non-linear and is optimal near 50%).
Rather than depending on the lights, measure voltage at your battery posts before hooking up a charger or turning on the converter. 12.1 volts represents about 50% capacity; 12.6 to 12.8 volts is fully charged.
2014 Rockwood A122 A-frame with dual golf cart batteries (232 AH each, 116 AH available)