A small point to keep in mind when looking at batteries may be in play here.
Are you familiar with what they call surface charge on batteries?
It can lead to missing what the actual battery is doing if we don't keep it in mind, so thought to mention it.
A battery is a slow moving chemical process and charging or discharge takes time that can't be rushed to far.
Different types do it at different rates but if we are speaking of lead acid, it takes hours to get from really run down back to fully charged. Like maybe 6 hours!
What we can miss if not alert, is when we look at a battery on charge and see the voltage from the charger.
It should always be higher than the normal battery can hold as that is what makes the current flow into the battery.
One way it is explained in schools is charging batteries and testing voltage is somewhat like pouring something thick like syrup in a hole in a barrel! Goofy sounding but works to explain some of the problems!
If we test voltage of a battery while charging, it may be something like looking at the syrup running in the hole of a barrel we are filling. It looks like the barrel is full as we see syrup, but if we looked a bit further down in the barrel we might find it nearly empty down where we can't see! Same with battery voltage!
When we test voltage, we can only see what is at the posts and if we are pouring power in, we see lots, even though the rest of the battery may be near empty!
But when we stop charging and test voltage too soon, the syrup/power has not had time to spread out all through the chemicals in the battery and we may not get the real idea of how well charged the battery until we wait for it to settle and become stable after we take the charge off.
This an trick us if we are not really alert. If you saw 13+, that will be the charge voltage as the normal really good new lead acid can only hold a charge of near 12.8 at best!
If you put a battery on charge for long enough to get it fully charged and just have it setting not connected to anything, you will first see the 13+ but it will soon go down to 12.8 or so, even if not connected to anything!
But if there is a problem with the battery, it may keep on going and get really down, like 10 volts if it is not in good enough shape to take and hold a charge!
Something that sounds like what you might have?
When driving for a 4 hours, the alternator should be connected to ALL the RV batteries and try to charge them to around 13+. So you get to a campsite and check voltage and you may see 13+ for a few hours until it settles? You plug in and see the 13+ from the shore power and converter feeding the coach batteries. All normal there, but if you unplug from shore power and charging stops, the voltage may go down really too fast if the batteries are not in good enough shape to actually take and hold that charge.
Kind of like the syrup and barrel, we can only look at the hole or post and the battery may actually only be half full down where we can't see!
A stop by an auto parts place where they put a load on a battery to test can be free and get some good info on the battery condition.
They can be so simple but still be so tricky, that we have to look really close at times.
Hope that's not a bunch of boring stuff you already knew but I have to keep reminding myself at times.
Good luck on the chase!