With gen on, the converter would be on making "12v" and running the 12v things and charging the battery if it needs charging.
Converters can go bad and go over their usual voltage outputs of say 14.4v in boost. The fridge has a DC limit of 15.4v. Perhaps the converter has pushed the 12v system voltage above the fuse limits.
You need a voltmeter to check things out. Meanwhile kill the converter's 120v circuit breaker while the gen is on. You should still get 12v (the proper values) from the battery before it runs down. Meanwhile you can find the problem. You can use a regular portable battery charger if the battery does run down.
