I cant see why you couldnt get one. The only stumbling block you might come into is you have to provide a valid reason on your FAC for requiring one. If you look back over the years when the LBR first appear on the market after the pistol ban. Most police forces were not happy issuing variations for them mainly because there were no competitions at the time for them. Then after a few years and plently of organised competitions there are no issues. As for FAC air there arnt any UK competitions that spring to mind.

You best way to do this without any hickups would be start a home office aprooved FAC club and arrange postal competition between a club abroad. Obviously you would require land suitable for that calibre range you intend to shoot and range insurance.
Speak to BASC, NRA and your local fireams lisencing team. They will provide you all the details required to open a club.

Good luck

Chris