Dominic,
To calculate your energy needs, there is a program called "HOMER" which is put out by the National Renewable Energy Lab (NREL). You can play with all the figures and calculations with it including your power generation, expected load and size of battery bank you'll need. Of course they stopped making the free one and now expect you to pay for it. They do have a web based version that's free, government at work.
HOMER software
From there you can see what your budget allows for.
I have to agree with what others have said - consider it an experiment and don't plan on relying on it as a full time power source. And make your life easier, get an alternator with an internal regulator. And the RPMs of the alternator is also important. They put out maximum power within an RPM range. In your car, assuming it's an ICE based system, your system voltage with always be around 13.5 - 14 volts when the engine is running - expect that with your setup.
The thing with doing this is that you'll want to feed the charge into a deep cycle battery and then draw it off for the inverter. Also, inverters have a range of voltage it will use, usually as high as somewhere around 13.5 volts. With the alternator whirring away, you may wind up over that high end voltage. And once the power generation is shut off, you have what potential you have stored in the batteries to power your devices.
A very important note is this: Wet cell deep cycle batteries will be ruined in a few months if you go below a 50% charge. Size your system so that you use no more than 30% of it's capacity when not generating power. So many people over the years have asked me about why their battery bank is toast after a few months. They don't realize that deposits build up on the plates when they're discharged. Sometimes putting an equalizing charge works, sometimes not. There is a fluid that I use in the cells that prevents much of that which is a commercial product. I've got one deep cycle from 2006 that still works fine using that.
Also, plan on everyone complaining because it will be loud. Like standing next to a lawn mower for hours and hours.
I've had some success with what you're talking about but still, you'll be fiddling with it a lot. I'd like to suggest that if you really want to succeed, assuming that you abandon that particular setup, get a small inverter generator. There is only really one company that makes the engines for them, Ducar. It's the box and features that each company puts around the engine. My Ryobi 1800 watt inverter generator runs for 15+ hours on a gallon of gas at around a 30% load. I use it to charge the battery bank (430 amp hours) during the day and power the tiny home.
The Ryobi I have works at 57 db and in the generator house it sits in, 20 db.