Once you've faced something there should be no runout at all. Since you do have some runout I would look at the spindle for the problem. Either the faceplate is rocking or the spindle bearings are loose or something. Might be you're measuring the difference between a fre-trning backplate and one that has cutting forces applied to it.
First thing I would do is look at the seating face on the spindle nose. Face the chuck seat and re-cut the chamfer so there's no dings or raised burrs on it, then re-mount the backplate and face it again. Check the results. If they are no different you will need to start looking at the spindle, find out what the problem is, fix it, then re-face the chuck seat. Don't attempt to turn the OD register if you make it undersize there's no going back.
Not wishing tostateask the obvious, but are you SURE it's the right one for the lathe?
His first post said there was no movement levering it about. It's quite mystifying.
Did you try clocking the spindle nose?
I'd start off by measuring the runout on the spindle itself. If this is dead on then I'd check the runout on various points across the diameter of faceplate.
If you make the cut it should have no runout as it should have cut concentric. If you remove and replace the faceplate you normally get any induced error in the mounting of the face plate.
It could be that the error is coming in because of movement in the cross or compound slides when cutting or measuring.
If you make a cut at a single point with a round nose tool to make a groove and then clock it does it still have the run out?
In theory a facing cut should not be a thou out as it's just cut from the same point relative to the spindle centreline.
Are you getting any chatter?
Also are you locking the compound slide the error could be from there. Have you tried a different position angle on the compound?
A dished backplate is of no consequence. The chuck will still mount up and run true. A domed one is a different matter.
I seem to recall reading somewhere that a lathe should be setup to turn slightly concave, even from new - which sounds contrary but there's probably a good reason that I've forgotten...