Jeff Layman wrote:
How about using a 13A fused spur with flex outlet? It might be possible
to take the flex to an extension socket (assuming it's allowed under
regulations). No plug needed.

+1

Any socket is going to be at least as deep as the pins on the plug, so
there's a limit to how slim you can be. Using a fused spur avoids the depth
of both the pins on the plug and the screw terminals inside the plug.

(Can you fit a fused spur inside a 16mm box?)

Theo