A 9000BTU air conditioner requires just 2.63KW and 12000BTU air conditioners requires 3.51KW. So I guess it will be ok for the rest of the house as well (lights and fridge only).
I think you should better ask an expert first as I heard air conditioners need many times more wattage to start working. in that case, it might not be able to power up.
An ac compressor may consume 45,000 watts of power while the motor is trying to start - TEN times more power than it uses while running.
The wattage I described are the starting and not the running wattage. Hense, it is covered. Anyway, I should know, I am in the air condtioning field since 1998
the wattage on my post says 45000 watts which makes it 45KW. what you described is 5.5KW and that's more than 8 times less. hence it's not covered at all