Thats a lot of voltage sag powering a DC fridge depending on the actual wattage it draws. Mine can go as high as 6.5 amps if I allow the compressor to run at 3500rpm. I only need 2000 rpm and there is a brief 55 watt start up surge and basically 32 watts running.
Either the battery is small, or sulfated or your fridge is drawing way more than 65 watts is my internet diagnosis.
---edit
Just saw a 35Ah battery in previous post.
See how long it lasts before it shuts down from teh low voltage cut out, so you know how long you can run it, assumnig you can keep the battery relatively healthy.
Keeping it healthy is all about the recharging, do not worry so much about teh 50% level, its not like 49% and BOOM!
Recharging that 35AH AGM should get about 12 amps until voltage rises to 14.5 to 14.9v, hold that voltage for 4 more hours. Amperage should taper to about 0.2 to 0.3 amps by the end. if you can recharge this battery like above I'd expect 200+ deep cycles running just the fridge until it shuts itself off on the lowest battery protection setting.
If your are going to expect 400+ cycles then do no draw it much below 50%.
The 50% rule is a bit nanny driven, especially on AGMs. Work the battery hard, just recharge it at the right amp rate to true full charge promptly and it will work well even discharged to 10.8v each cycle