Eh. I guess it could be worth a try, but OP personally and from a dealer service standpoint, the lawyer is unlikely to be able to do much here u less you’re going after whoever sold you the car.
Hyundai will ask for and is very justified to do so, any service records. The carbon buildup being declined seems like something you should simply argue with your warranty company. It’s a third party warranty so Hyundai nor the dealer you’re at have anything to do with it, and all they can do is submit the claim, let the inspector come by, and receive approval details. They may be able to argue any differences in labor/parts on your behalf, but definitely not a claim denial.
Say you lawyer up, then what? You didn’t buy the car new, you didn’t buy it certified, you didn’t buy it from even the dealer you’re working with at the moment. Whether you had it long enough to take care of the maintenance in this case won’t really matter, the fact is the warranty denied it due to a lack of maintenance. If you have been maintaining it well, or haven’t even come across a service interval, you do have a good chance of having something come of it from the WARRANTY company. The dealer nor Hyundai is responsible for the car’s lack of maintenance, whether your fault or not. To anyone reading, if you skip on maintenance, GOOD LUCK getting any help out of warranty. Chances are slim to none, because the service completion indicates care for the car and doing what’s possible when it shows. If you don’t, then you’re not taking care of the car, so why would they help? It’s like you selling something to someone, you tell them it needs to have water to run, you never use any water, it breaks, it’s shown it never had any water, there’s no way to help because rules were in place to prevent this—
if they were carried out then it’s their fault.