Disputing with the card issuer probably would go not much further than the supplier demonstrating, "here's the e-ticket that was issued at the time of purchase." If there's a ticket number, the service was "delivered."
The issue seems to have been resolved where it belongs--with AC. They can put all the fine print in they like, but when they charge extra fees for using portals other than the web/app, they are probably going to find little support when they try to rely on, "not our fault if our site doesn't work," language.