connecting to shore power

Maybe the problem is not at all in the battery and power source? Maybe the problem is in the voltmeter, did you check it with another voltmeter?

Sounds to me like a mismatch between the battery management system that should be built in to the battery and the rv’s converter programming. And absolutely you need to get it fixed pronto before it destroys the batteries. Lithium batteries are very expensive, and while they will last long enough in service to pay for themselves vs lead-acid batteries, the cost to replace ones killed by a bad charger will take your breath away.

I'd deal with this promptly or face having the LI's fail.

14.9 sounds high for typical lithium. Maybe the BMS is cutting off power.

What is the make and model number of the charging system? Could it be set up with an automatic equalization mode? I don't know of any standard RV system that goes above 14.6 volts.

What kind of chargeing system do you to charge via 120 volt shore power? It may not be of the correct type for lithium batteries.