Norcold N611 acting odd, affected by interior coach lights

Your converter could be the problem. You need good 12-volts DC to refer. The converter or shore power is not providing proper power so the refer is cycling to propane.
Verify voltage to the refer with a good meter.
Try to listen for clicking while testing voltage. This needs to be done in outside access to refer.
It is possible that circuit for the refer is failing. I somewhat do not think it is failing.