A couple of thoughts...
The chuck will go on more than one way relative to its bolts. Try each of the positions until you find the one that gives minimal runout. FWIW, I marked each postion on my lathe's spindle and each position on each chuck after doing this with just some automatic centerpunch pips: 1, 2, and 3 pips. Now I can always reinstall the chucks for minimal runout and in the same position.
With that said, I doubt it will fix your whole runout problem, or even likely the majority of it.
These things are kind of hard to diagnose remotely. You're going to need to get out your DTI and start sweeping until you find what is out. As I mention, my chuck's outer ground surface is WAY out when it is running true. Ignore the outer surface. Instead, start from the spindle and work your way out until you sweep the indicator and find the problem.
For example, remove the chuck and its backplate and sweep the spindle's spigot that mates with your backplate. Then check the backplate, then you are back to the chuck.
How did you attack the chuck to its backplate? In my case I used a shrink fit, which is very tight and has no clearance. If you just bolted yours up, perhaps there is some leeway there that's working against you. Make sure there is no chip caught anywhere too.
Let us know what else you learn. Hard to tell at this stage just what went wrong.
Best,
BW
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