I bought this device to use in my RV. It has run great on the 12V Cig lighter plug.

Oh yes . well theoretically they provide 2 Amp adapter bundled so lets assume at max stress device sucks 2 Amp current .
A car battery is normally 50 Amps . so you would get 25 hours of backup ..
sometimes it may fall apart because i have seen many batteries discharging and their voltage drops to 11.7 V

Oh yes another thing. if possible provide power near to ap. Shorter PoE cable lesser voltage limit your device will run on