All charging goes to the loads first, then the batteries get anything left over. If the loads take it all and want even more, that has to now come from the battery.
That is why RV converters are set to 13.6v as "normal" for while camping, and only go to 13.2v "storage" for battery maintenance when not camping and there are no other 12v loads.
The OP's 1972 converter needs to be checked for actual voltage no load and then when under the expected 12v loads, to see if it can hold the loaded voltage up enough so the battery does not have any draw on it.
The loads should be connected on the other side of the battery from that old converter so the battery will act as a filter for any "dirty 12v" that could ruin the fridge controls and any other circuit boards in the various 12v loads.
BTW, did we discover whether that "15 amps" is its 120v rating for the AC breaker panel or indeed for its DC output rating?