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Since you are adding additional cooling capacity, perhaps a smaller and or more energy efficient air conditioner may also work for you; with added benefit that it would draw less power after startup as well.
I've researched air conditioner's start-up current for a different reason (to run off inverter), and think it "may" be possible to run two smaller 11,000 BTU/hr Power Savers off a 4 kW generator and or 30-Amp service (depending on other loads). These units are marketed as ideal for smaller generators (perhaps they are referring to the common Honda 2000???).
As an example, many Coleman 13,500 or 15,000 ACs momentarily draw 63 AMPs at startup (Lock Rotor Amps), but the 11,000 PS has an LRA rating of only 45.6. The difference is around 17 Amps, which "may" be enough for the 4 kW generator to keep powering a first unit already running while the second unit starts up. Unfortunately it's not quite as simple as adding the Amps, but I think it has a good chance of working.
Once running these smaller ACs are rated at 9.9 Amps under standard conditions (higher for dessert conditions), which should allow two units to run off 30-Amp service while providing 22,000 BTU/hr combined.
I don't know if Coleman ACs are any good so I'm not recommending this particular brand. I'm just suggesting that it may be worth looking at ACs that draw less current while providing your cooling needs.
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