Generally the choice is between plugged in with the batteries on against unplugged with the batteries off.
Wet lead acid batteries self discharge at only about 5% per month. So if not plugged in, the batteries are off and (importantly) nobody has hooked up something directly to the batteries, you're good with nothing to worry about for a few months.
The other choice leaves more to worry about. The inverter/charger should be "smart" with a "float charge" rate, it has to be working correctly, the batteries have to be healthy for the "float" charge to work, the isolator/BIRD/Big Boy system has to be working and capable of maintaining both battery banks. You also have to service the batteries with distilled water about once a month.