Do your other 120 VAC appliances work with the generator? It seems odd that just the air conditioner would not switch over on a 30A system—is there maybe a separate breaker (on the generator) for it, or a separate transfer switch, or something?
Replacement of the transfer switch shouldn't be too terribly difficult; in concept it's basically a straightforward substitution. I'd guess the worst of the difficulties would be minor headaches like getting access, having wires that aren't the right length, discovering that the new transfer switch is 1/2" bigger or smaller than the old one or has knockouts in inconvenient spots, etc.