As I stated above Royal mail will do so (surface mail only) subject to their size limit, PF48 will not export guns or gun parts so to try is against their rules & if found it will be seized /destroyed.
AFAIK no other UK carrier will knowingly carry an airgun for a member of the public, let alone export it.
The parcel will now probably require a customs declaration form so lying about the contents has gone up a notch or two