yeah in English saying please my hat is more like saying making my hat happy.....
I would agree with the previous comment. In English, it is more common to put the please at the end so as not to sound so formal.
this reminds me of an episode of Donald and this robot.... your hat sir, please.. (takes hat) DONALD: PLEASE MY HAT sir! give it back...
Please my hat sounds like you want someone to entertain or make it happy.
I know it's not literal, but I answered, "Give me my hat, please.", which is essentially what it means. I also agree with the comment that many English speakers would say (in a literal translation), "My hat, please."