I've run a 5000 BTU AC off my Kipor 2000W inverter-based generator set for full power. But the generator would occasionally trip-out trying to start a 6500 BTU AC.

There's a video on Youtube of a guy starting and running a 5000BTU AC off a 900W Harbor Freight generator, but I have strong concerns about the reliability of that setup.

Rule of thumb is that an air conditioner draws 2.5X the power during start-up as when running. I calculate 3X to allow a safety margin. Meters are not always a reliable indicator of surge power requirements because they don't update the display fast enough and don't have a peak load function.

As for emissions, always run the generator outside with the exhaust facing away from the house. Keep windows, doors and vents (including the fresh air intake on your AC) closed in the close vicinity of the generator. Putting the generator on the opposite side of the house from your AC unit is not necessary, and in hot weather you're more likely to want those windows in other parts of the house open for breezes.