I agree that their request is unreasonable, and I'm sorry for the stress it's causing you. I can't really justify their behavior on a rational basis (teens, right?). And obviously I don't know your child, and I don't know you, but as a trans person who went through a similar phase, maybe consider what they might be thinking if you want to understand why they might be behaving like this?
For one thing, they almost certainly know your opinions about the persistence of trans identity and online echo chambers of gender identity. If (real if) you're hostile to the idea that they could really be trans, it's possible they feel hostile to the idea that you could really be making a good faith best effort to support them, which is what this sounds like to me: defensiveness. I had a very similar dynamic with my parents for several years. We came out of it understanding each other a lot better, but it was certainly rough in the thick of it. I really hope you can find a way to resolve the tension.