Sounds like you are on the trail of the air leak.
If I run the inverter as we drive in depletes the house batteries commonly down to 11.5 volts after a couple of hours running the refrigerator. I would have thought that the engine charger would keep up with this draw.
It depends ...I presume your Journey has the large 4-door Norcold 1200 series refrigerator. It does take a good bit of power. I can run mine on inverter while driving, but need to be aware to shut it down or switch to LP if we stop anywhere for more than a few minutes. My MH is set up to charge ALL batteries whether on engine, generator, or shore power.
The primary question I cannot answer is, does your engine alternator charge the house batteries in addition to the chassis batteries? If it doesn't, there is your cause. If it does, the problem may be that the batteries are in poor condition. If they are somewhat sulphated, they won't take/hold a charge well and will go down quickly. Do the batteries work well in other situations? If your charger/inverter has an equalize function, and the batteries are not too far gone, that can give the batteries new life. The first time I equalized mine they were 2+ years old and not doing well at all. Equalizing made a huge difference in their performance. I normally equalize my batteries a couple of times a year (I am a fulltimer ...if I were part time I would probably do it only once a year).