Gene,
Yes you can run the fridge from an inverter.
No, you may not be able replace all the energy drawn from the house battery to power the fridge from the alternator. If you find a way to do so please post about it.
If the battery bank is big enough the failure to support the energy use is a non issue. I used to have 875 amp-hours of battery bank. That meant the losses didn't make much difference to me, when I was part time, because the solar system would recharge the bank fully during storage times between trips.
I'm a propane miser and run my RV nearly 100% electrically. Propane use for the year was $34. Further I'm a generator hater, and ran mine less that 50 hours in the past 12 months.