The problem with the Store/Use switch is while in Store mode there is still some battery draw from the LP Leak detector so after a couple of months the batteries could be drawn down to unsafe levels.
The problem with the OEM charger/converter is its not a 3 stage charger that monitors the state of the battery. It will continually charge the battery causing the water to eventually boil out if kept on charge. To solve this issue an converter upgrade is needed such as the Progressive Dynamics PD4655.
You could turn the the Use/Store switch on for a week then off for a week. The trouble is on shore power the LED is light for Use and Store modes so hard to tell unless you put a multimeter on the battery to see if its at 12.x or 13.6v from the charger.
If you know the batteries are fully charged you could put the Use/Store to Store and just disconnect the negative cable and leave the trailer on shore power to power 12v
on the trailer.