Here is the info on exporting a firearm from the US.
Q: Does a licensee need an export license to export a firearm?
The GCA does not require export licenses. However, most firearms and ammunition must be exported in accordance with the provisions of the Arms Export Control Act of 1976. Regulations implementing this Act generally require a license to be obtained from the Directorate of Defense Trade Controls, Department of State, PM/DDTC, SA-1, Room 1200, 2401 E St., N.W., Washington, DC 20037; (202) 663-1282.
I would call the number and see exactly what it is going to take or find a US dealer that has already done it. I would also try to find a Canadian dealer who imports or has received firearms from the US.