I was wondering if it was possible to create a custom overlay on my own pet and then transfer that pet to another user with the overlay still applied?
This is probably a ticket question since I'm not sure if it's intentional or still possible, but yes, that at least used to be possible :) The "ownership" of the overlay doesn't transfer though. I adopted out two pets that had custom overlays applied, and they still show up with the overlays on them, but those overlays are listed on my account (though I haven't asked the current owners if they show up at all for them or anything). Filing a ticket would help clarify that :)
If you're looking to specifically gift the pet + overlay to someone, it might be better to send them the files and CSC to upload it themselves, just in case