Many converters do not react well to not having a battery connected. The battery acts as a filter and load to the converter. If this is a trailer that you tow around, be aware that without the battery in place you will have NO breakaway (emergency ) braking system. Although most 12V DC systems use fuses for protection, you could possibly have a 12V breaker protection in your unit (not clear in your description). If it is truly a breaker, it may be time to change it out as they do start to blow more easily if blown too often.
Have you tried changing that particular bulb, it may be defective and shorting out at times. Unusual failure mode but not unheard of!!! Could also be the lamp fixture and switch acting up when switched on. Try disconnecting the connecting wires in the back of the fixture after examining the wiring, and see what happens. You mentioned "above the slide out" and "flickering" and since those connecting wires may move I suspect there may be a hot wire with worn/cracked insulation or loose in that area.
We may be talking of an AC breaker here as well as it would power the converter itself and that may have been affected by the no battery condition over the years and just now starting to show, or, it is on it's way out.
Your outlets are a completely different system, they are 120V AC (not 12VDC) from the shore power as you pointed out. It (or they) will have c/b protection. The lights are part of the 12V (and battery/converter output) system.
