I've read over the replies from a lot of the others, and they are suggesting that to check the charger, but I don't see where anyone really specifically suggested that you check the battery voltage, and what it should be. Also, the battery going bad may depend on if you have a battery switch, an A/B/C/All to allow different batteries to be in use. That could make different batteries go bad, especially if using the charger off the engine.
I noticed that someone referenced the manual for your charger. The question I would have, is if the temperature sensors from the charger are actually installed on the batteries. Anytime the batteries are changed, the temperature sensors should be reinstalled on the battery. Including the seasonally if they're pulled out during the winter and put back in the summer.
The other thing that you would want to do is to check the voltage across Each battery under different conditions. With the engine running and the Charger off, use a digital voltmeter, available at many electronics houses, inexpensive ones at Harbor freight, RadioShack, or hardware stores. You need to set the voltmeter to about the 20 V scale, and measure the voltage across each battery ( red to the plus, black to the minus to Posts), especially on the battery that is switched in (a/b/c). The engine alternator charging system should show about 14.2 V to the battery that is being charged, engine running. And that should never go above that!
When your shore charger is hooked up the voltage may go up to about The same 14.2 level, but often times the Chargers will back off to about 13.2 V, just to maintain the trickle charger part of the trip should be to measure the voltage at different times to make sure that the voltage does not go over there, or stay real high for a long period of time.
Mike