Thought of one other idea: I've heard that controllers also have low voltage cutoff. I guess that means it could be the either the BMS OR the controller shutting down. The fact that you can get this to reset by turning off the controller makes me think it's the part making the decision and not the BMS. Maybe others know if a BMS that's entered the shutdown state takes time to reset?

Your idea to try different batteries is sound, if not expensive. I hate to see you spend money on heavy, relatively short-life SLAs just for a test. If I had this problem I'd scavenge 3 batteries from my lawn mower, ATV and snowmobile. Heck even my better half's car battery could be missing for a while, just to test of course.