This is the unit...

Essentially, it is a pair of relays, of which each can operate as N.O. or N.C.
Currently, because the battery is disabled, it is set to be N.O. for each leg of 120V from main panel (and goes to the power sense on the generator). The control for the coil is connected to one of the grid lines. If power from grid is lost, the connection relay opens, and generator runs. When grid is back, it switches back to grid/PV. It is set this way, only because the battery is not operational.
Before, they told me it was set to N.C., but I'm not sure exactly how they had it wired. I
think they had it set up so that the dry contact on the BUI would close, if battery went below 20% and no grid?, which would cause the generator to run and power the panel and recharge the battery. I don't know what battery threshold would close the contact and go back to battery/PV/grid.
Nor do I know how it handles the edge case where the battery is dead for some reason and the grid is out. The contact would be closed, but no voltage at the generator sense wires, so generator would run. But, then there would be voltage at the generator sense lines (from the generator, through the main panel) and I would think the generator would stop. Unless the inverter could boot up, determine that battery was low, and have enough power to signal to close the dry contact on the BUI? I'm not sure the inverter could startup that quickly (unless they signal the dry contact first thing?). All conjecture, of course.
I didn't get to talk to the installer about that edge case and how it would be handled - and whether or not my understand of the wiring was correct (other than I know they run it N.C.).
Right now, I'm just trying to get the RMA'ed battery so that they'll come out and connect it back up the way they had previously done. In isntaller is not very responsive (frustrates me to no end). After emailing/calling them about 10 times over two weeks, I finally found out Friday afternoon, that LG is processing the RMA and will be sending tracking info next week (I was told by the installer that LG said they have switched over to a new system and were slow in processing RMAs as a result).
So, two weeks have passed and LG is just starting the process to send a battery. A much slower response than last time, when the battery was delivered in about 2.5 weeks (of course it took the installer a month to come out to install

).