Sump pomp problem

I assume it's a main (120VAC) + backup (12VDC) sump pump system.
I think the main one is the green, and the backup is the white (StormPro) unit.
(see https://www.sumppumpsdirect.com/manuals/SP2100DC_Manual.pdf for manual, and https://www.youtube.com/watch?v=nUl1kYhhCD0 for a similar unit)

I have couple questions:

Q: am I correct that this system only has one floating switch?

Q: am I correct that this system should be working even if 12VDC is not connected, for testing purpose?

The way I test it is:
1. the black unit has 2 cables coming out of it, and I connect them to the labeled "float" on the back of the black power control box, as shown below (I've tried switching the black/white cable, neither work)

( the front of box is like

)
2. and then connect the power cables (two of them) coming out of the green and the black units to the power outlet
3. manually lift the floating switch to the top
but then there's no response at all. Am I correct that at least one thing is broken, and it can be: a. the green sump pump, b. the black unit, c. or the black box ?

Q: the only way to test the backup unit (white one) is to connect it to the 12VDC battery right?

I've heard the beeping sound from the system since mid last year (yes I was lazy to wait this long..), which indicates the battery is dying. I've also use multi-meter to measure the voltage of the battery, which seem to confirm that. So, the only way to test the backup unit is to connect it to a good 12VDC battery, right? Will my benchtop power supply work for this case?

(some pictures of the battery

)