I don't think that even the folks at PayP would accept such a ridiculous note as proof of authenticity. And honestly, I'm pretty sure that you can buy that notepaper (with the little Louis Vuitton in the corner) as an insert for an agenda. I almost 100% positive that they used to sell it, because I seem to remember having it in my agenda 5+ years ago.
Hang in there, OP. Get your beautiful bag back from crazy buyer & resell to someone that will appreciate it! I would also recommend getting your own letter of authenticity and mentioning to the buyer that you will be 1) getting your own letter and 2) will need the name of her "friend" as you will be bringing this letter to a LV boutique to confirm that it was truly prepared by a LV associate. Then tell her that she's welcome to submit her "proof" to PayP or she can return your (secretly marked) bag to you & will be refunded once you confirm that the bag she returns is YOURS. Honestly, what is with these people???