I "almost" did similar back in the seventies. Had a tight spot. Don't remember "how" I did this "It might have been" Proto 3/8 drive allen bits used to have an allen set screw so you could pull the bit and change the thing. May have been one of those or a junk socket. Anyhow I think I brazed a bolt the correct size to screw into the rear of the chuck, into the drive. Was not nearly as short as yours!!1
