On a filmholder the depth is the distance from the outside of the filmholder, that rests agaist the camera, to the septum against which the film rests. On the ground glass back, you want to measure the equivalent distance to the ground surface of the ground glass. Normally the ground glass will be aboout 0.007 inch closer to allow for the thickness of the film.
The standard depth for 4x5 filmholders is 5.0 mm = 0.197 inches, which implies the ground glass should have a register of 0.190 inches = 4.83 mm. The ANSI filmholder standard specifies a tolerance of 0.007 inch. The table
http://www.mppusers.freeuk.com/registers.htm gives the register for the MKIII as 0.200 inch = 5.08 mm (avoiding metric, they were...). So, right off, if a MKIII is exactly on its target register, its just past the ANSI standard tolerance: 0.010 versus 0.007 inches
The ANSI spec is reasonable. The equation for depth of focus is +/- C N, where C is the diameter of the circle of confusion and N is the f-number. A standard value for C for 4x5 is 0.1 mm; supposing f11 as the fastest taking aperture, gives +/- 1.1 mm = 0.04 in. This is larger than both the ANSI tolerance and the likely register difference between the MKIII and the standard.
Also, it is possible that Lee's camera (which he doesn't even have yet) has been modified or upgraded in the many years since it was made -- hopefully for the better, possibly for the worse. I suggest a careful photographic test of the focusing accuracy, or a measurement of the register with a depth micrometer, or both, rather than relying too much on the table of the original specs.
I haven't seen a MKIII, but most cameras have the position stops in front of the ground glass, so adding shims will move the ground glass away from the lens, increasing the register, which is the wrong direction.