I guess I should have gone through the numbers instead of posting what I did. I was trying to do the whole "give a man a fish / teach a man to fish" thing. I apologize....
Anyway, I agree with family_wagon that there is a lot of wiggle room, and I agree with the statements about the alternator, but with a caveat that I will note in just a sec.
So, working backwards from the load, which is a 4A @ 120V load (or 480W), through the inverter (say, at 80% efficiency just to be conservative), would make 480W/0.8, or 600W on the DC side. At 12V, 600W is 50A. Therefore, if the DC source is able to put out 50A @ 12V, then the load should run fine.
Since the alternator is able to put out 145A at full load, and the AC only needs 50A through the inverter, then everything should run just fine - at full load....
But if the engine is running at idle (and this is the caveat), then the alternator's output may very will be less than full load. Therefore, the OP should check the alternator's specs to make sure it can generate a minimum of 50A at whatever the engine's idle speed is.
Just remember that these numbers are conservative, but they are also rough. This means that they should be nearly "worst-case" but not at "worst-case".