Mr. Mike,
I would leave it plugged in to keep the batteries topped off. Also, I would get AGM batteries when your new ones need replacing.
I would leave the 'auto generator' active in case you lose power as you don't want to ruin your batteries assuming that you have some type of ventilation if stored indoors.
Where I store my coach, it has lost power a few times and my generator kicked on. It's a huge warehouse type facility and it was not a problem with the genny running. We have 3 lithium ion batteries that can go down to a 30% charge then the genny will kick on. We have four 3,500 watt inverters to keep everything going and charged.
Safe travels,
MM.