I would guess that your starting 12 volt batteries are the problem.
They lead an easy life and should last a very long time (I got almost 10 years out of mine) providing they meet the 950 CCA each specification. If they do not meet the 950 CCA the service life is much shorter.
I am assuming that your coach is outside in sunny California and your solar panel is functioning. Once all your battery switches are off, your solar charging system is hard wired to the battery banks and is controlled thru the fuse box that you refer to. This is your Battery Control Center and contains a relay which is likely what you hear clicking on shore power or generator power.
The BCC will direct all charging between the 2 battery banks, once one is charged it will via the relay begin charging the other bank.
If your 12 volts batteries are shot, this is not possible...hence the clicking as the BCC keeps switching from one bank to the other.
That is my theory.
(For those who have newer coaches, these vintage Eagles do not give you the specifics on float charge levels etc, they are much more simple than the new units that have sophisticated energy management systems etc.)
If you do need to replace your starting batteries, go with some 1050 CCA rated batteries that are available and widely used in farm and construction equipment.
They are available in the same size that your starting battery trays were designed for.
I got mine from Advanced Auto Parts. They were Carquest brand.