Don't rush to judgement on the alternator if your coach battery is not under duress.
It's typical for the BCC to have the vehicle charging system charge the house batteries, and disconnect from the vehicle system when the alternator isn't charging (running), and thus prevent the house from drawing on the coach battery.
If the 6V batteries are not matched, you get parasitic losses. Matched in this case means similar capacities, of which age and condition will play a part. (you aren't getting truly "matched" batteries off-the-shelf, but that'll be close enough usually).
If you have one battery that is compromised in some fashion, then 2 things occur - using the batteries causes the "better" battery to bear an excess of the load, and the fact that the bad battery isn't coming up to the same level of charge necessary to pull it's share means the charging system voltage continues to exceed the combined peaked voltage of the 2 6ers.
That's why batteries are replaced in pairs. A reasonable matching of capacities. I don't know enough about your overall coach, BCC, etc. to go much further, but if it was me, I would first isolate the 2 systems by disconnecting the coach charging system from the house batteries. If your BCC is working properly, that just means unplugging the coach, and not running the engine. Depending on your model, you may be able to just turn off your inverter/converter, but in any case, just unplugging will also do that very effectively.
Now check your voltages resting. House pair, and coach. Run some 12v stuff in the coach for a bit. A few lights, whatever. Or watch TV and pull those puppies down a bit.
Don't kill them or run them flat.
Now go do some measuring. The coach battery should be happy and ballpark 13v +/- or so.
Precision isn't necessary - you just want to know that the coach bat is happy. Your twin 6ers should be noticably lower. How much? Doesn't matter, but enough that you know you drew them down using them. Note that value. Disconnect the 2 6 volt batteries. Measure them individually. Are they about even, or is one noticeably lower than the other one? If yes, investigate the lower one, for when you again plug in, the current delivered to these, in series, is going to be high to recover the "lower" battery, and is going to cause hate and discontent not only because it's discharged, but also with it's teammate as they are joined at the hip, so to speak.
If, however, they appear substantially equal (they won't be exact; even new ones off the shelf are not precision), then you will want to dig in and understand your BCC a bit, and also determine if they get boiling mad off the coach's charger when plugged in to shore power.
If ya want, run your headlights a bit after all this, pull your coach batt down (
a few tenths max; it's a starting battery).
Reconnect your house batts you just checked.
Fire up the engine. Hang out at the BCC. Possibly you will hear the solenoid link the house batts to the charging system shortly after fire up. Check the house batts (connected up in series again as usual). Voltage should be up if your vehicle is charging them. Cool. Ditto the coach battery. If the house batteries were pulled down during your test, you will absolutely have voltage fluctuations in the vehicle system.
Here's the thing about the alternator fluctuations - your house batteries can and will take on a good amperage load when under duress and getting charged by the vehicle. As heavy duty as the cabling is, when you get up into many 10s of amps, there is some cable losses, and the alternator/charging system is far from precision. If you have a battery that is really straining the system, it can easily pull down the whole shebang and be visible as fluctuations. Check out the voltages at each battery during this charge up test as denniscw mentioned. I think the results of that are more relevant immediately after loading them down.
Have you done tests using the converter/charger? Which unit is it? Freedom 458?
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