Okay, let's step back a minute. The 13-14 V charging vs 12.4 V at the batteries is puzzling, yes, but it seems to be precise.
When the engine is running, the batteries are seeing 13-14V, charging, yes? When you stop them, are the batteries now at 12.4v, or higher? The reason I ask is because previously you were taking readings at the LVD and it was 12.3V, and I'm wondering if at that time the batteries were at 12.3V?
I mean, it seems like on charging for alternator you read charging voltage, and when not alternator charging you read normal batteries voltage. If there is an issue with some ground, I'd expect to see something lower that normal battery voltage somewhere along the route to the distribution panel.
Can you please go measure using the following process?
This is a photo of a converter/charger panal (not mine, just pulled off web), should be close to what yours looks like. The BATT POS + line is top left with red and black wires attached. To the right of that, after the 2 30A fuses, is CONV POS+ from the converter with a black wire, and the posts NEG- BATT and CONV with white wires, which touch each other.
1) While alternator is charging, and shore is not connected (and generator not running):
a) what do you have at batteries?
b) What do you have at LVD input?
c) What do you have at LVD output?
d) what do you have at the fuse panel (distribution panel) when measuring between BATT POS+ and NEG- BATT posts?
e) what do you have at the fuse panel (distribution panel) when measuring between CONV POS+ and NEG- CONV posts?
2) Turn engine off (alternator not charging now), and shore is not connected (and generator not running):
a) what do you have at batteries?
b) What do you have at LVD input?
c) What do you have at LVD output?
d) what do you have at the fuse panel (distribution panel) when measuring between BATT POS+ and NEG- BATT posts?
e) what do you have at the fuse panel (distribution panel) when measuring between CONV POS+ and NEG- CONV posts?
f) what do you have at batteries?
This is just a double check in case you quickly lose .1V of surface charge while measuring everything (I don't think it happens that fast, but I've never actually checked). E.g. if batteries are 12.5, then measuring b-e you get some at 12.4, I want to see if batteries have settled to 12.4 during measuring.
3) Engine off (alternator not charging), and shore IS connected and should be charging (and generator not running):
a) what do you have at batteries?
b) What do you have at LVD input?
c) What do you have at LVD output?
d) what do you have at the fuse panel (distribution panel) when measuring between BATT POS+ and NEG- BATT posts?
e) what do you have at the fuse panel (distribution panel) when measuring between CONV POS+ and NEG- CONV posts?
Do all this and let us know.