The start assist / hard start kits are very inexpensive. Google for examples. You should address the sag issue. Forget about the generation part of it when / if you complain to the electric company. They have a load device that replaces the meter temporarily, and the measure the drop when the load is applied. You'll want your inverter locked out while they are playing so as not to "help". Get someone to measure the voltage at the input to your main breaker with a fast recording meter. Look at how low it goes when you put the ~ 100 amp start load from your AC on it. If you go from say
240 to 215 volts for the instant, that would be a 10% drop - way too much for 1/2 load on a 200 amp service. The comment about 268 volts at night is interesting also. I'm thinking transformer issue. The primary side - at most utilities - is regulated within 2.5%. Mention the range you are seeing, and they'll likely put a logger out there, followed by a transformer replacement.Is your transformer pole mounted or on the ground? If pole mounted, see if you can figure out how many houses are on it. Then look at the number on the side of it. 10, 15, 20, 25, 37.5, 50, etc. That is it's continuous capacity in Kva. 200 amps at 240 volts is 48 Kva. It's assumed that no-one is going to consume 200 amps at 240 continuously. The realistic number is something like 10% of that. The real issue is the total load on that transformer.