We could never prove my theory, but what I suspected happened was that there was an improperly-crimped plug on the molex-style connector to the second alternator: it had pulled free one time when I unplugged it (to prevent further over-charging), and appeared to have no bends in the copper wire strands where it should have been crimped onto that little "bullet" end.
Not really conclusive, but it did have the appearance of not being crimped properly. My further suspicion was that that particular wire may have fed the feedback circuit that determined how much amperage to put out to the coach batteries, hence the constant overcharge, but again, that's just an educated guess. We ended up with 3 out of 4 batteries totally fried and one that had only about 24% of its capacity left in it.
This was when we were in Oregon; I had been in touch with both RT and the outfit that designed the extra charging system (Nations Electric, Point Girardeau, MO). We couldn't stay long enough in one spot to have the system repaired, so we just dealt with it (plugged in every night) until we headed home to NH. We stopped at RT in Ontario on the way back, and they replaced essentially everything, even gave us a car to use for the day.
Replaced: All four batteries; second alternator ("underhood" generator); alternator wiring harness; and, IIRC, some of the heavy wiring back to the two batteries in the rear.
I do not recall whether the additional alternator had an external regulator or not, but it could have. I just didn't happen to see one.
I'd suggest you do two things:
1) Contact Gary (I think) Nations at Nations Electric and talk to him; I found him to be very straightforward and helpful.
2) Take a close look at that molex plug's wiring where it connects to the alternator.
I suspect the 12.4V you measured was simply the batteries' standing/resting voltage. Sounds like the charging system's doing nothing.
Hope this helps, and let us know what you eventually find.
EDIT: I'm making the assumption, of course, that your system is similar to mine...which certainly may not be the case. According to research I did back in 2015, Nations Electric, a family-owned business, has been designing, fabricating, and in some cases installing, high performance electric systems for marine, automotive competition, and RV use for a number if years. I believe the arrangement was that they designed the system used in our Agile and shipped the parts as a "kit" to RT in Ontario for on-site installation.