Note: I've been doing this for the last 7 years =I run the inverter-directly from the 12v battery. ***During the Hurricane outages - The inverter runs my full size refrigerator, Two square floor fans and a few other small electrical items in the house
I would imagine you can do that and if you had like four or five 12 volt batteries you might even have more headroom I don't know how often it charges the full size refrigerator is about a 7 amp draw I think on startup and then just sitting there maybe 7 amps or something along those lines the floor fans pretty much no issue and the small appliances I'm assuming phone chargers and that sort of thing shouldn't really be a problem to put that in perspective to run my property here if I wanted to do such a thing when the power is out would take a about 20,000 Watts 23,000 somewhere right in there I could get by with essentials with around 10,000 but if I'm going to get $10,000 I might as well get closer to the 20 or 23 and be able to run it all problem is I don't want gas or propane it has to be diesel Diesel will be here forever the military runs on diesel Diesel's never going anywhere.
we have a propane full house generator. power isn't too reliable around her, even in the best weather
P=IV, and the power it needs is the power it needs. 7A at 120V is 70A at 12V. A continuous draw of 70A would certainly be a problem for a 45 AH battery. But it isn't 7A, it is usually 3 to 6 amps. And it only draws current when it is running (or a little when the door is open for the bulb). https://www.researchgate.net/figure/Power-consumption-of-an-old-fridge-and-an-old-freezer-in-site-S5-during-a-time-interval_fig4_275974446 So let's say a fridge uses 4.5A (average of that range), with a 25% duty cycle (this is probably optimistic). That is equivalent to a continuous 1.125A at 120V, which is only 11.25A at 12V. Seems like the 12V would only run for about 4 hours before it was depleted. The OP never said the fridge was plugged in the whole time though, nor that the car was turned off, just that the inverter was attached directly to the 12V. The car's inverter can definitely supply more than 11.25A at 12V, so as long as there is still gas in the tank, it would keep the fridge happy.
Between our tri-fuel 220v Honda 5,500kW (100% duty cycle) genset in the garage, & the 3,500kW diesel Onan built into the MB motorhome ... bring it on! (plus, we live up on a hill ... so when/if the city's sewage pump dies, we won't have to go "dig a hole" for several days) .