Good to know that you are a Ham operator. Helps me have a better understanding of your skill sets.
The fact that the 4-6 volt batteries are connected in a series / parallel configuration, one bad cell in one battery will drain the batteries.
There is a bypass relay in the unit, this allows you to have AC power to the 2 circuits running from the inverter when it is turned off.
Allot of things going on, so I created a list to work from so all know what and where tests have been made. I know you have run some of them and a yes, no, a voltage or other response to the item list will sure help get everyone on the same page:
1- You replaced all 4-Good starting base line.
2- The AC and refrigerator worked until the voltage dropped to 10 volts. This is a voltage level that is below the needed voltage for them to work. Normal!
3- The batteries are charging when the engine is running. That is a good start!
4- Your coach is wired with a 50 Amp service, this means you have 2-120 volt feeds. Often referred to as L1 and L2. The Charger / Inverter in your coach requires 120 volts at 30 amps.and is connected to ether the L1 or L2. Coach builders choice!!
5- So check the shore power cable for any issues with the Neutral and Ground circuits in the cable ends, running a continuity test; between the ends of the cable for all 4 circuits.
6- Check the circuits from the J box at the coach for the shore power connection to the AC breaker panel.
7- Check and see that all the connections are tight at the J box and the breaker panel.
8- Locate the breaker that feeds the 458. look for good tight connections
9- with power going to the inverter, there are 2- 120 volt 20amp breakers that power 2 circuits inside the coach. These supply power to the Microwave and one other circuit for the front TV and a couple of other outlets. These outlets should have 120 volts when the inverter is running, with no shore power or the generator is not running.
10- turn off the inverter portion of the charger. With AC coming from shore power or the generator, read the Dc voltage across the 4-6 volt battery pairs.
You should should read close 14.5 volts DC. and the Red LED on the controller should be light; but only for a short time. Then the Yellow LED should come on. Then after a period of time can be an Hr. or more. the green LED will light.
NOTE. any interruption in the charging cycle will cause the 458 to go through the complete cycle again.
11- If you do not read the 14 plus voltage at the batteries when the charger is running. Disconnect one set of the 6-volt batteries from ground. This leaves you with just 2-6 volt cells in series, rerun the test again. Run the same test with the other 2-6 volt batteries grounded and the ground from the first set disconnected.
What readings do you get ? the same or different?
12- Get the same reading on all the tests, then disconnect the common ground for all 4-6 volt batteries and read the charger voltage. What is that reading.
13- What is the condition of the wire and connections for the thermocouple connecting the 458 to the batteries?
14- Post the results and we will go over the battery wiring configurations for the 4 -6 volt batteries and the 2 chassis batteries.
Questions, are you using a copy of the owners manual that I supplied the Link for ?
Do you have a Power Driven power cord reel in the coach ?