running an air conditioner on 250 feet of cord

250', #10 wire, 12 amps, drops about 6.5 volts. If the supply is good you should have no trouble. Voltage booster has its place but will draw more amps to make up for lost voltage. This could have your 12 amp draw too close to 15 amp to avoid tripping a breaker. True 20a should work either way. Continuous wire would be preferred over five EHD extension cords cobbled together.

I do wonder how long your solar/converter/inverter would hold up running the air.