If you try something like this, you wouldn't want to use car batteries.
You need to use deep cycle batteries instead.
Car batteries aren't designed to be discharged to the degree that an application like this would do.
You also would need a large amount of battery capacity to run an electric fridge, lights and a small heater for a few days.
I have a little bit of personal experience with this.
In my case, I have a cabin in the woods and there's no electricity.
I didn't want a generator making all that noise whenever I wanted electric power.
I wanted to have basic lights, music (an old laptop computer with a music library on it),a small stereo amp, radio, the ability to charge battery packs for cordless tools, etc.
I even wanted to run one of those large coolers that run off 12 volts.
More importantly, I wanted this stuff to be available 24/7.
After doing this for a few summer vacations, I arrived at a solution that works OK, and doesn't require a large battery capacity.
I have two Trojan L16H 6volt 420 amp hour deep cycle batteries wired in series for 12volts.
These batteries weigh about 120 pounds each.
I have an 1100 watt Exeltech pure sine wave inverter connected to them.
I went with pure sine wave so there wouldn't be any issues with plugging things into it that don't like non-pure sign wave AC.
I use 7 watt fluorescent screw in light bulb replacements for lights.
I found that even the small electric coolers pull between 5-10 amps at 12 volts.
This and especially if somebody else brought one along, really drew the batteries down when run 24/7.
So I use an old propane powered refrigerator/freezer out of an old RV to take my refrigeration off of the electric circuit.
It will run 24/7 for about a week and a half on a full 20 pound (gas grill sized) tank.
If I need heat, I use a kerosene heater, or burn wood in a small stove.
For cooking, I have a big tank of bottle gas outside.
So now I can run my system for 2 days between recharges if I don't waste energy. (I usually charge every day so I don't discharge the batteries as far, thus extending their life. They 'aint cheap LOL )
It consists of 7 watt fluorescent lights, an old laptop computer, an old Marantz 1060 30 watt per channel amp, a fluorescent shop light for an outdoor light when needed at night.
I try to charge any cordless tool battery packs when I do my main batery charge.
So now you ask, what does it take to fully charge these 2 Trojan L16H 6volt 420 amp hour deep cycle batteries wired in series?
I have a Honda 6.5hp motor, with a modified 65 or 70 amp (I forget which) Delco alternator.
IT'S MODIFIED SO THAT IT DOESN'T USE AN INTERNAL VOLTAGE REGULATOR, BUT INSTEAD WIRED TO ALLOW ME TO CONTROL FIELD CURRENT EXTERNALLY.
MY NEIGHBORHOOD AUTO ELECTRIC SHOP DID THIS FOR ME.
IT'S IMPORTANT THAT YOU DO THIS, BECAUSE IF YOU JUST HOOK UP AN ALTERNATOR WITH THE STANDARD VOLTAGE REGULATOR TO DEEP CYCLE BATTERIES, YOU RISK EXPLODING THE BATTERIES WITH THE HIGH CURRENT THE ALTERNATOR WILL PRODUCE (BECAUSE IT'S NOT BEGINNING THE CHARGE REGULATED BY CURRENT INSTEAD OF VOLTAGE), OR YOU WILL BURN UP THE ALTERNATOR.
Just for this pair of batteries, I need to begin the charge cycle with 40-50 amps. (plus a little more to account for any current being drawn in the cabin from the inverter, which for each 1 amp of AC, it pulls about 10 amps from the battery bank)
It runs at the level of amps until the battery voltage reaches 14.8 volts, then the current is backed down gradually as the batteries charge to keep the charging voltage at 14.8 volts.
So you can see that it's easy to run the alternator at close to it's rated capacity pretty easy.
The good thing is, it's not in a hot engine compartment, so that helps.
So say I would want to double this battery bank?
Already the small Delco alternator won't have enough capacity to properly charge the battery bank, as I would then need 80-100 amps.
And the 6.5hp motor would be getting close to not being big enough.
I'm planning on going to a battery bank of four L16H batteries in the near future, so I recently upgraded my alternator to a Leece-Neville 165amp externally regulated alternator.
These are pretty expensive.
So getting back to your idea of using your generator to charge a battery bank during the time you have it running?
Imagine the size of AC generator it would take to run an AC battery charger big enough to charge these?
Anyway, MY POINT in writing all this is to make the point that what you're wanting to do is a bigger project than you would think.
I know when I started down this road back in 2000, I had no idea it would take as much as it has, just to have 420 amp hours @ 12volts capacity.
I have thought of messing with some solar panels in the future, but haven't yet, because the cabin is in thick woods, and there's not enough sun.