Hi Corndog,
Congratulations on the new North Trail and welcome to the Heartland Owners Forum. This is a great place to get answers as there's a friendly and helpful bunch of owners who are quick to reach out when you need help.
Most of your lights are probably running off the 12V DC system, which will get their power by way of the Power Converter which uses some 110V AC current. The Converter will consume some of your 20 amps running lights and keeping the battery charged, but you may be able to get away with also running the A/C unit. If the compressor happens to start up when the Power Converter hits its maximum, you could trip a breaker.
Two other considerations: you don't want an undersized extension cord, especially for a long run, and often "20 amp" outlets are really 15 amps, which might not be enough.