If your Danhard is like mine, a combination of AC 120 and DC 12v, one needs shore power or a generator. If on shore power with a charger charging the batteries, no you do not have to run the engine. Need at least 20 amp circuit breaker though in my experience. If trying to run it by battery alone, yes you need to turn on the inverter. No idea how your second alternator is wired, but I would think it is always on or there is a manual switch or an automatic switch.
I have run my generator and Danhard while driving a short distance.
How big is your battery capacity? It would be possible to do it by battery and inverter, if one had a large enough battery capacity and double alternators. The AC draw is too much for mine without shore power or generator. I have not heard of anyone running a Danhard by batteries alone.
Hopefully someone with more knowledge will respond.
2008 GMC 3500 Quigley Weldtec 4x4 Savana SMB