Jani, could I suggest you call or email ATF. They are very easy to deal with and you'll get your information straight from the horse's mouth. I called them from Afghanistan once when I was sending some Martini-Henry's back to the USA via APO.
https://www.atf.gov/contactThe ATF is the wrong place to go for this. I went down this road a long way with them because most of the really don't know. The State Department is the correct place for information. If you are unsure, email one of your state's senators. They can be sure helpful, and they (their people) know their stuff.
Rules for shipping parts and guns and barrels on particular, are pretty simple but easy to get wrong. BATF is domestic only.