You are going to have to cut a hole in the side of the cabinet to get the cables through, why not cut this so that it is directly over the screw that is inaccessible. It looks like you would be able to reach through such a hole, even with a normal length driver to get the screw out. The hole will be covered by the surface mount socket, or will be the whole you cut for a backbox if you install a back box into the side of the cabinet. I don't like the idea of cutting any more out of the cabinet than necessary, but you can give the customer the choice of surface or recessed.
Once you have the old socket off, I would go with your idea of extending the cables in the existing backbox, feed them out through a blanking plate with a 20mm hole and a grommet, and through the hole you made in the cabinet.
You will need the hole in the cabinet to be big enough (50-60mm dia?) so you can get a screw back into the blanking plate working through the hole - think keyhole surgery! Needle nose pliers, a screwdriver with an attachment to hold the screw firmly onto the blade, or a dab of superglue on the end of the screwdriver; all might work.
For the best possible job, the gap between the cabinet really needs covering somehow. If time were no object, I'd get a bit of 18mm ply, primed and painted to match the cabinet, and attach it to the side of the cabinet from inside. Just make the fixings obvious and leave a note for the next poor spark on how to access the original back box.