First thing is the batteries while being charged get up to a higher voltage than what they rest at when fully charged and not being charged.
So 14.6 if that is what the charger gets them to and then after they stay at that until fully charged (amps have tapered right down, and hydrometer says SG is up to where it belongs when fully charged--say 1.285 or whatever then they sit at 12.75 or whatever. Takes time for that voltage to fall and until it gets down to 12.75 say they are passing through 13 on the way down, that is called a "surface charge" the amount it is over "resting" voltage.
Second thing is the battery can look good from its voltage before any load is on it, then once loaded, the voltage will drop somewhat depending on the amps being drawn by the load. eg if batts are at 12.6 and the furnace comes on, it might be showing 12.4 or whatever.
BUT if the furnace comes on and voltage drops to 12.0, then you know the batts are "collapsing under load" so out comes the Visa card. :(
Meanwhile, the converter (with 120v input) with no battery in the RV is supposed to show 13.6ish and run the 12v things. It too might collapse under load or have a weird voltage that is too high like 18v or whatever.
With the WFCO if the battery is connected and the battery is low enough so the battery voltage stays under 13.2v (the trigger amount) for its initial spike, then the WFCO should kick into 14.4v. Many don't for various reasons ever see that low of an initial spike so they stay at 13.6 and only do a slow charge on the battery.
So yes, two separate things to look at before connecting the battery, then see what happens.
Your overnight drop in battery voltage could still be from an unknown mystery draw. If you have an ammeter or battery monitor that shows amps, you can see if there is an amps draw right away where it might not show as a voltage sag until some time has passed.
Some mystery draws that have been found include 12v tank heaters, somebody pulled the emergency brake switch on a trailer, TV antenna booster, 12v lights in underside compartments left on with doors closed so you can't see the light is on, 7-pin connected and the tow vehicle is drawing from the RV battery (happens with Chevs--no isolator like a Ford has)
If going to get batteries load tested they are supposed to be full when the test is done.