So if your house battery voltage in the Firefly view does not increase when you start the engine, then the Li-Bim, BIM has not connected the chassis battery to the house battery or the chassis' alternator is not working.
There is also a "boost" function (as I recall) on the front dash which can manually short the chassis battery to the house battery. You might play with this also to see how the chassis battery and alternator can influence the house battery.
Others are also right. The magnum power converter and the DC/AC inverter have nothing to do with the alternator charging.
Oh, one more thing. As I'm lead to believe (although I have not looked for it) there is a circuit breaker under the driver's seat on the wire that connects the two batteries. If this had tripped, then you would see chassis battery charging but no house battery charging.
Good luck!