Hi Tom--
This situation sounds a little tricky since from your reports and indications it sounds like the batteries are only failing to charge while plugged in to shore power. Since that is unusual, it would be really helpful to verify that with some test readings with a voltmeter (which you indicate you have with you). Depending on the state of charge of the batteries, it's a little difficult to predict exactly what voltage readings you should expect, but checking the house battery voltage for instance with the RV plugged into shore power, and then with the unit off shore power but with generator running, should confirm if you're showing a higher voltage (charge voltage) across the batteries with the generator on than when plugged in.
Please note that the Ford engine alternator does not in any way use the converter/charger or TempAssure module to charge the house batteries. That is a totally independent circuit using only the 12-volt vehicle charging system to charge the house batteries (as well as the Ford battery). However, you ARE correct that both charging while plugged into shore power or by running your Onan generator each use the convertor/charger to convert 120-volt AC to 12-volt DC power to charge the house batteries--so if you confirm absolutely that the converter/charger is working properly on generator, that should mean that the converter/charger unit is OK, and the problem would seem to be that it is not being properly activated while plugged in to shore. That is why DonC mentioned the transfer switch--since that unit is what changes the input to your converter/charger from shore power to generator when you fire up the generator, and it could be responsible for you not receiving shore power to your converter. That is highly unlikely though, since the transfer switch switches ALL A/C power, and if it's not giving you shore A/C to your converter, it's likely not giving you shore A/C at all---over a 4-day stay, you would almost certainly notice if you weren't actually getting shore power and the only A/C power is what you had available from your inverter. As he mentioned, the microwave is an easy test for that, since it is not powered by inverter, so would not operate if the transfer switch was not giving you shore A/C--bottom line, though, is that the transfer switch wouldn't seem to be a likely culprit. I suppose that one possible answer could be that the shore power you were using was so far out of tolerance (extremely low voltage) that the converter/charger couldn't provide an adequate battery charging voltage from it, but if you have any type of EMS or other protection on your A/C power that would have caught it. Also in that case the situation should correct when you left that particular hook-up.
Another consideration to factor in is that you lost a set of batteries during your recent storage period apparently because they were overcharged and boiled dry from an extended period of being plugged in to shore power. That would seem to indicate that the batteries were being charged on shore power at that point. These various reasons are why I suggest trying to confirm that what appears to be the case of no charging under shore power only is in fact what's happening. If that proves to be the case, there would certainly be some cause/explanation for it--but it likely would require some direct troubleshooting on your system to determine the problem and fix it. If you DO stop by Elkhart, consider having them install the Trimetric TM2030RV Battery Monitor while you're there. That is a nominal cost option on factory builds and they should have units on hand. It consists of the monitor/meter and a shunt that installs near the batteries. With the TM2030, you have a constant readout in amps of current flowing into (+) or out of (-) your house batteries, exact battery voltage, and percent of charge (0 to 100%) of the batteries; it's a great system that eliminates guesswork and enables you to know exactly what's going on with your house battery bank.
All the best, Mike