The chance to wear ANY hat in the game just makes this item too valuable. Perhaps there's a small pool of specific hats you can choose from, like craftable hats, or just older hats. It could be a good solution to retiring crates from the game. Other than that, maybe instead of picking from one of any hat in the game, it can pick from one of the hats in your inventory. That'd make it less of a must-have for players.

My second piece to add is the way you obtain this item. Regardless of how the hat behaves, it seems more like a tool than a definitive cosmetic item. As such, it'd be disappointing to have it be unboxed from a limited-time crate, and then not be as easily available later on. Perhaps it can be an achievement item that gives new players a chance to wear a random (not so valuable) hat without having to pay for anything. Lets F2Ps wear hats, though they have little control over how they look. A gibus in disguise, if you will. If its mechanics make it more valuable, however, perhaps it can be a secondary tool drop like name tags or unusualifiers.

TL;DR Either make it pick from a pool of less valuable hats or make it pick a hat from your inventory. If the case is the former, let it be an achievement item. If the case is the latter, let it be a rare secondary drop like tools.

PS: Secondary style that doesn't make it random because that's a lovely looking model <3