If you feel the need to get very precise, you can "face" the nut on an engine lathe, or vertical mill if you prefer.
Since you know the TPI is 16 - you know that each revolution = .063" advancement. With six "flats" on the nut, you also know that the advancement of each one = .0105".
So, as an example, if you estimate you need about a half a flat advancement for a proper fit, you need to face off .0105/2" = .0052". Any machinest, home schooled or otherwise should be able to do this with out problem.
I do this on all my "fit ups" to insure I get the fit I want instead of what the suppliers give me. Also, if needed, the depth of the nut itself can be cut down quite a bit, if doing so helps with the wheel pant install.
If you like the idea of using shims to "adjust" the fit for future wear, I would suggest the following:
Extremely thin shims (a few "thou") have the tendency to ball up in a wad when they are installed or are too fragile to remain intact during their service life. So, undercut the axle nut to allow a thicker shim to be installed initially, to give the desired fit. Then as conditions warrant, change the shim size (but still thick enough) as needed. From the Aviation Division of McMaster-Carr, for 1 1/4" diameter axles, their shim thickness from .031" on down are less than or equal to .005" - should be close enough.
YMMV - and probably will.
HFS