My good old Barbour Border is getting a bit funky inside.

Isn't this merely a "feature" of a well worn Barbour? ;-)

How does one clean the tartan fabric if the jacket cannot be put in the
washer?

On a more helpful note how about some sort of "mousse" carpet type cleaner?

Or I've had quite some success cleaning grubby unremovable upholstery fabric
with a steam cleaner. Old toweling fitted over the team outlet to soak up
the scunge lifted by the steam. That may or may not make a great deal of
sense :-) If you have a suitable steamer then maybe you'll know what I mean.

Henry