What's the story with the cranky POE with the PowerBox Pro? Is this unit not for 12V battery operation?

The spec says "DC jack input Voltage 12-57 V, PoE in input Voltage 12-57 V " so you are very close to minimum power, I always use 24V .

use 24V.
If you need to power from 12v battery, use a 12V->24V DC DC adapter.

use 24V.
If you need to power from 12v battery, use a 12V->24V DC DC adapter.

I wouldn't use a DC converter because of inverter efficency which maybe 85-90%, much easier and more power efficient is to put second 12v battery in series for 24v.

24V input is best, but if existing charging system is 12V i would prefer a DC DC.