I didn't know you had a 1960
Overlander. Good for you. I almost bought a 59 Overlander (photo below) but backed out due to the single axle. I just felt it was too much length for a single axle, sway, sway, sway... Maybe your Overlander is a perfect boondocker. Like my old 66 Trade Wind, it doesn't have many amenities that draw power; a bed, and chair, a table, and a potty.
Collecting vintage Airstreams is a fun hobby, but more expensive than gardening.
Car starting batteries are completely different design from marine deep cycle batteries. I can't recommend a car battery as they won't hold a charge very long in your trailer. If you boondock in your 34, buy two batteries and stay an extra night or two. Buy a solar panel and stay an extra 3 or 4 nights. Buy a generator and run the AC and charge the batteries. We boondocked for two nights and found the batteries about half gone. We missed our electricity to charge computer batteries. Too attached to our coal fired electricity I guess.