I'm near Sittingbourne, and shoot at Kent Woodsmen. Takes about 45 minutes to get there from Sittingbourne. No indoor shooting there, and they are predominantly Field Target, though do have a plinking range. FT course can be shot as HFT if you are into that.

ETL just the other side of the Dartford tunnel is very similar, but have slightly better facilities (container with electricity used as club house where hot drinks can be had)

As others have said, there is a club on the outskirts of Dover which is really glorified plinking from small wooden huts. I've heard about the shooting at the Black Lion in Gillingham/Chatham, but have no idea what format it takes or who you would need to contact.

HTH