Question about a motorhome's dash air conditioner

I think our motorhome may be a bit light on rooftop air conditioning when we get into the summer.

My question is this. Will running the dash air conditioner with the engine off but the coach plugged in to shore power work without draining the battery? With the engine off, I know the blowers work, but will the AC also cool from just battery power, without the engine running? And it's seemed to me in the past that the battery up front gets recharged from shore power - is that also true? If it is, I should be able to supplement my cooling without resulting in a dead battery. Thanks.