Two battery chargers (Alternator/chassis battery and inverter/charger/converter with generator) can both work together charging the house battery bank. If generator is supplying all house circuits, it is likely supplying the charger as well. The charger supplying the highest voltage will supply the most current for charging.
It is certainly possible that some EMS is stopping the charge due to its internal programing, but it is usually not necessary and so is not programmed in. Designers can do just about anything these days.
Trouble shoot by following the voltages. Charging voltages start at the static state of charge (SoC). They slowly rise to the preset charger voltage.
Alternator may possibly supply 14.4 volts. This will raise chassis battery terminal voltage to 14.4 volts. At some point the system that connects the house and chassis batteries should connect and raise house battery terminal voltage to 14.4 as well. Deeply discharged batteries may take a while to reach the 14.4 volts.
Inverter/charger may also supply 14.4 volts or at other times 13.6 volts.
Fully charging lead acid batteries takes 14 to 18 hours. A good fast charger can bring a deeply discharged battery up to 80% SoC in 4 hours. It still takes another 10 hours for a full clean charge. The batteries have limited ability to absorb current. That ability decreases to nearly nothing as the battery approaches full charge. It takes a long time to finish. It cannot be pushed faster.
You can force more current, but the batteries won't absorb it.
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How does the Lead Acid Battery Work?
How do battery chargers work
How to Charge and When to Charge?
"Static" State of Charge (SoC)
After 3 hours rest.