I’ve handled warranty claims for a major tire manufacturer - and if the 2015 date on the belt is correct, the OP doesn’t have much of a case. He would need to prove that some other component failed that should have been replaced - and since this occurred 2 years after the belt was installed, the OP just doesn’t have enough favorable facts to get a judgment.
Further, timing belts are made out of different materials than tires and don’t age in the same way. It is very likely that if the OP sued, the dealer would get someone to testify that it wasn’t a defective belt, nor is 2 years too old for a timing belt to be considered new - AND - that all the pulleys and the tensioner showed no signs of needing to be replaced at the time of the installation of the belt.
Sorry, but that’s the way I see it.