You should have a multimeter, especially as a full-timer. Check the voltage at the battery lugs by the converter's output-- sometimes they are on the fuse panel. Turn off shore power and you should get battery voltage of 12.x. Turn on shore power (to supply the converter) and you should see 13.x or more if it goes into bulk at 14.x.
With no meter, you can check with your DC lights--same thing only lights will be brighter at the higher converter voltage than at battery voltage.
You idiot lights battery monitor is actually a voltmeter. With battery only it might show a green or yellow light. With the converter voltage at 13.x (way above a full battery voltage of 12.7) the monitor will say "full" battery--even if the battery itself is dead.
The converter could be plugged into a receptacle, which is dead from being on a popped GFCI circuit, which the fridge's receptacle (outside in the back) might also be on. You can check those receptacles with your multimeter for 120v or plug in a 120v lamp that is known to work.
The power surge might have killed the converter with too much 120v. There is a 120v fuse inside the converter if you are comfortable taking the lid off and poking around. It is right by where the 120v wires are connected to the circuit board.