Rick T, sorry I missed your battery install post. Here is some more helpful information about that one.
I have no idea what the first owner was trying to accomplish. His switch simply defeats the internal transfer switch purpose on the inverter. But, if the inverter is still powered to charge the batteries the internal transfer switch will still be active unless the inverter is turned off.
in any case.
For keeping your batteries watered do as many of us have been doing. This trick goes all the way back to Edison and his RR power station maintenance in NY. He had his men put mineral oil in each cell of the batteries. I do this on my wet cells. Works like a champ.
Mineral oil has no reaction at all to the battery. It acts like a valve seal. All batteries gas but the bubbling releases corrosive moisture and that is what reduces the water level in the cells and corrodes the terminals and connections. The mineral oil lets the gasses escape but not the water. Many of us have gone years between water adds and our corrosion is very manageable. Any drugstore carries mineral oil as a laxative and it is very inexpensive.
Using it is not rocket science. After the cells are topped with water simply add a ounce or so to each cell. It just has to cover the surface area and doesn't need to be thick.
Where are you reading the 13+/- jumping as opposed to the 14V reading? Most coach monitoring panels don't agree with the inverter remote reading. I have no clear answer why.
New batteries want to be cycled a few time to get them stable. I suggest running dry dock
a few time over the next week or so. Being summer this may be a problem.
Here is one solution that may work: If your coach has a generator auto start system that is sensitive to the state of the batteries you are golden. If you have cool night temperatures you can run your roof fan to pull cool air for sleeping. If you have a space heater, plug it in to load the inverter. (Of course your shore power is off for this time.) When the batteries sag to the generator trigger point, it will start and recharge your battery bank.
Repeat this cycle for a few nights in a row and then test the battery voltages.
If you have a standard RV refrigerator running on DC this may be the current fluctuation.
Hope things improve for you. Many happy trails.