Low battery and failing ESA makes me think ...
I once had been charging my battery. As I don't have a typical BMW charger, and as I am reluctant of charging the battery while it remains connected to the bike, I always disconnect the (plus pole of the) battery. Now after the battery was fully charged, I hadn't reconnected the battery to the bike right away, but only after a couple of months (it was the first winter I didn't use it any longer for going to work).
When I then reconnected the battery to the bike after that long time, my ESA didn't work any more. It looked as if my ESA switch button was broken : pressing it resulted in no ESA reaction whatsoever. The dealer saw it, didn't look very carefully at the diagnostics, made the shortcut-assumption that it was the button : "they use to break quite often in these models, I'll order one, you won't have to pay anything because that's a flaw of this bike, I'll put it under warranty".
Some days later, after the piece had arrived, the switch was replaced, the bike was started ... just to show the same behaviour. The dealer starts looking at the diagnostics more carefully now, and sees that the ESA had lost its calibration point (or whatever it was called). So he runs the job to recalibrate the ESA ... and there the ESA was working as expected again !
Now to make this long story short : if the battery has been flat enough, or flat for enough a time, I would certainly not exclude the possibility that the ESA has "just" lost its calibration point.