The character is a hedonist. "the ethical theory that pleasure (in the sense of the satisfaction of desires) is the highest good and proper aim of human life." Such a person would not stop being a hedonist just because Tiamat needed to be stopped. The character would do both, or that character isn't really a hedonist. Hedonism is the foundation of that person's life. It's not going away.
Well, first of all I think people are more complex and less consistent than you give them credit for here. Just because someone is a hedonist doesn’t mean they will always act out of hedonistic motivation. A person can and will set such things aside when they have more pressing concerns. A hedonistic character could still recognize that, in light of the goings-on with the dragon cult, it is in their best interest to temporarily prioritize stopping the cult over spending money on carnal experiences.
The claim wasn't that it was the only way. The claim was that a hedonist would spend money on hedonism, which is true. The character would ALSO engage in free experiences. He would do whatever was necessary.
Would he necessarily spend money on such things when there’s a world to be saved though? Maybe he would, maybe he wouldn’t. People are complex and often inconsistent. At any rate, just because a player decides that their hedonistic character would save their money for things that could help them defeat the Tiamat cult doesn’t mean that character is nothing but a player avatar.
There are a lot of good suggestions here regarding ways for characters to spend money. I think its just as important, though, to consider the other half of the equation: an excess of gold can be avoided just as readily by reducing income as by adding expenses.