I've heard that the fence will take away a few feet of space for the dogs so I'de rather install a wall a few feet into

 
are you thinking of an INVISIBLE fence? I'm suggesting an ELECTRIC fence, the kind people use to make horse and cattle corrals-- a strand of hot wire run across the problem area. Dog can't dig without touching the hot wire. Touching the hot wire is very unpleasant. You can buy a simple setup for quite cheaply.