There are two kinds of old-time RV converters before they came out with the modern switch mode type that don't need a battery connected to act as a filter.
Some of the 1990s types do need the battery hooked up to act as a filter for their "dirty" DC outputs and some don't AFAIK, but I have no clue which.
However, the old converters, whichever type, are so low in amps output that they can't really cook your new batts in only a few days, so IMO you are in no danger for the two days before your trip that you are plugged in to run the fridge and the converter is on the batts.
For longer term Float, you can get a Battery minder sort of portable charger ( maybe $30) to use instead of the converter if it is a concern.
