Lead acid batteries do not fare well under full discharge. While a proper circuit design would prevent this and remove the need to disconnect the battery lead (only need to do one!) you can hedge your bets by unplugging the unit and removing one battery lead.
I will be gone all summer in DC, so what is the best thing to do with the UPS for 10 weeks? Is it best to just leave it plugged in, should I just unplug from the wall, or should I unplug the battery as well? It is one of the conventional lead-acid ones from APC in case in matters.