Things may have changed, but it used to be you could not fly AA to London and transfer your gun to BA. That language is no longer on the AA website, but that does not means anything. You will need to call AA and see if they will make the transfer. I would suggest flying BA to London and then BA onwards. On BA you call them, and give them the brand , serial number and caliber. Several weeks before you fly and the confirm about three days before you depart. At the airport you will fill out the paperwork and keep a copy with you for your return flight back home. It’s very easy, but expensive.
