What run out are you measuring?
Surely, what is important is what run out, radially and axially does the chuck impose on work held in it.
Having the chuck body moving axially is of lesser importance, unless used as a datum.
Don't expect even a new 3 jaw chuck to hold work concentric to better than 0.75 – 0.125 mm.
If you think about the clearances,;scroll to body, scroll to each jaw, each jaw to it's slot, (in both planes ) that is pretty good.
Could it be that the chuck is not mounted properly on the 2Mt arbor, and that is what is causing the axial run out?
If the chuck axis is at an angle to that of the 2MT arbor there will be axial movement and eccentricity of both chuck body and work held in it.
Some more measurements may help find where the eroor lies.
It could just be indifferent, or incomplete fitting, or burrs.
Howard