Do be aware that there are actually two wattage ratings that must be considered: the running watts of everything you plan to run at the same time, and the startup/surge watts when things first kick on. Now an electric resistance heater, and the 'fridge on electric falls into this category, is going to have the same surge as running wattage (well, close enough you can ignore the difference). But anything with a motor, and that means your AC, which actually has TWO motors, will have a surge draw 2-3 times the running wattage.
And some generator makers call their models based upon the surge, and some based upon the running wattage. For example, I have two generators; one is a Generac 5500 and the other is an ETQ 1800. The former is 5500 watts continuous/6850 watts surge; the latter is 1600 watts continuous/1800 watts surge. So you really need to know both ratings in order to pick the right generator.
As for not tripping the 15 amp house circuit with both the AC and the 'fridge running might be a tribute to the circuit breaker more than anything. If we assume the 'fridge heater was actually running when you did your test (which is not a sure thing) and the surge when you turned on the AC didn't knock off the breaker AT HOME, that may or may not carry over to in the boonies, because there is one more draw you need to bear in mind: the converter and the status of your house battery.
If the battery was fully charged when you were home, the converter won't make much difference, but if the battery needs charging, the converter will try to charge it, and thus draw perhaps 500 to 700 watts to do so.
If you search here, you will find many people bragging that their AC runs fine on a Honda 2000. You will also find some of them returning later to say scratch that; it works only at home when the house battery is fully charged.
So the simple truth is that a 2000 watt generator (surge, that is) will run some ACs, but it is very marginal at best. A 2400 watt generator (surge) will work fine most of the time, but only right up until somebody tries to microwave some popcorn . . . .
If all you want to do is run a 1600 watt space heater while charging the house battery and/or running the 'fridge, a 2000 watt (surge) generator won't cut it, you'll need that 2400 watt unit for sure. If you want to make sure you can power the space heater AND the 'fridge AND charge the batteries, the 3,000 watt (surge) generator is probably your best bet. And if you want to allow for that popcorn, too, better go for a 4,000 watt unit.
But heck, skip the space heater, and ONLY charge the batteries, then a 1,000 watt generator will do fine.