The unit has one job in each state of the power-flow display at the top of the page. While the grid is on, a 12V supply feeds a current-limited buck converter, and the UPS board splits that power between the lights and the battery. When the grid drops, the UPS board switches to the pack with no interruption. When the pack runs low, the BMS opens the circuit before the cells can be over-discharged.
The same unit already powers the routers in my own home on a 3S2P pack. Early testing found that a drained pack pulled over 60W through the UPS board's input diodes. That finding shaped the current-limited charging design below.