Your LP leak detector draws power even in Store mode and will run your batteries down after a couple of months. I always disconnect the negative battery cable while in storage. I rewired the LP Leak detector to the other side of the relay switch so I should have no draw when in store mode but I still disconnect the negative battery cable in storage.
Well cell batteries usually need a charge close to 14.8v then ramp down as they approach 80% then it takes a long time to get to 100% as chargers will ramp down the amps. With 13.8v with the OEM charger the batteries probably don't ever get fully charged. When the shore power is disconnected the voltage will ramp down to a steady state. My Interstates (grp27) seem to settle down to 12.6 then will come down to 12.4v and stay for there for a long time. Yours are probably grp24 which have less capacity. I always measure at the batteries. I have a cheapo battery meter in a 12v
socket and depending what is turned on; lights, fans etc it will always read lower than the batteries. I'm assuming the battery meter on the monitor panel is similar.
My Use/Store switch doesn't have click on/off. Its a self centering switch. The light comes on when pressing one way and off the other but the switch returns to center. When on shore power the light is always on so you can't tell if the batteries have been disconnected. If on shore and and the switch is on Store then the batteries could've discharged more than expected.
You might want to check the battery voltage while the tow vehicle is connected to see if they are getting a charge. You should see north of 12.8v. My trailer, preowned, had the charge line disconnected from the batteries and they never got charged while towing. Fixed that.