American Cap Badge on British Cap? Possible Eagle Squadron Cap?

[Carthin]

I purchased this hat a week ago and cant find any similar examples. At first I thought it was some sort of cottage industry manufactured hat for American GIs, but then I remembered about the Eagle Squadron and did some research on their patches. The booth I bought this hat from had a few ES patches for sale that I didn't recognize till I started researching. But it also does not match any ES hats I have been able to find pictures of.

It also could be something completely different.

So any help would be appreciated.

Thank you for your time,
Carthin