Fridge requires 12V DC in order to function on propane or electric.
You are losing the DC to fridge----lack of display, intermittent interior light, propane failing. electric nt working etc
Your battery may be good but the DC connection at terminal on backside of fridge in outside compartment may be loose, DC fuse connection on lower circuit board could be loose, loss of ground (12V DC Neg), could be harness between lower & upper boards and if installed in a Motorhome could be issue with the 'ignition interlock' for fridge doors
Could even be a flacky thermistor......your model will stop functioning if thermistor fails or has connection issues
Do you know how to use a Voltmeter?
Lots of tests/checks can be preformed .......just need to follow DC circuits and find where it stops.
Start with pg 18 (terminal block) and then pg 20---28 (various wiring diagrams depending on which 1350 model you have)