You would not need the battery in the circuit unless your converter needs one to act as a capacitor. Test it by turning off the battery disconnect switch while hooked to shore power. Your converter should continue to power everything 12V.
2014 F-250 6.7 with Timbrens
2015 Voltage 3605
Depending on the converter you have. The battery acts as a filter to even out the 12 volt power going to the various 12 volt devices. If yours is a fairly modern one it won't hurt to run without the battery for short periods of time. I wouldn't want to do it long term, that being more than a day at a time.
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2015 Coleman CM16FBS(traded) 2016 Concord 300DS
2015 Ford Fusion Hybrid following along
The connected battery acts as a damper, of sorts, to soften the output of the converter. Without it, the converter will jump up and down in output voltage depending on the need sensed. When you turn on a lamp, a TV, or the fan motor kicks in, the converter jumps to respond instantly. These "jumps" are seen as surges, and can age bulbs, and damage electronics. Sort of like tapping a water cooler tank, or directly from the city pressure.