The dino sprint hat. (Two star sprint hat)
True, but Tim was cut. The definition of "Time Lord" may have been changed. Then again, he is frozen in Queen Vanessa's manor, so that might also mean something other than a tiny reference.
So, would that mean she spied on people? That does explain why she has each different decorated room for chapters.
Just a question, if Hat Kid's art gallery has characters from earth, would that mean she went to earth already? That might extend the theory farther.
Theorists: this is actually true-
First, you need to know how the time stop hat works. Luckily, co-op mode is a thing. If Bow Kid uses the hat ability, Hat Kid gets slowed down and Bow Kid is normal. However, when both of them activate it, they both aren't slowed down. The same applies backwards. This can mean 2 things.
When a Time Lord slows down time, it affects other Time Lords, which also supports Bow Kid is a Time Lord
Hat Kid has knowledge of how to abuse time, which is why she knows how to craft it, and control it, meaning she spread the knowledge to Bow Kid.
I do not know if this logic applies to Online Party, as that would be chaotic, but that is it's own thing. However, the game does say Hat Kid is an alien so your theory can be true, it is just a bit more complex, or I'm overlooking it.
(Replying to your comment, her hat probably isn't one hat with multiple abilities, because she has to make every hat herself still.)
There are 2 versions of chapter 2?