Im not sure as why you went the route you did?
If this was me, If the chuck didnt feel right, I would have sent it back , after inspection.
Lets start from scratch here. You bought a new chuck, and new back plate. Assuming from the same seller, not that it matters. The back plate would be the first thing to fit up on your lathe. About all you should be required to do is take a skim cut to ensure it true. Second, cut the register step to fit the chuck. Both these operations on your lathe spindle. This would make both operations as accurate as your lathe is.
In some cases, you may be required to put the bolt pattern on the back plate to mount up. After all three of these things are done, mount it up, and check for run out. Now after indicating , you have run out issues, it can only be in a few places.
First thing would be a bad collet, it is possible. Id pull the collet,and indicate the ID of the chuck it self. Narrow it down, collet, or the chuck!
Providing you have trued the back plate as mentioned above, and indicated it to prove your work is accurate, the only places to check are the collets, and or the chuck its self. In either case, ship it back. I would not hang the chuck off your 4 jaw like that, and do any cutting. I have seen guys do this, only to learn later the run out is in the collet, or debree in the assembly. From what I understand this chuck/ back plate/ and collets are new. The only thing you should be cutting is the back plate, thats un avoidable. Past that ship the items that cause issues back.
Charley, just another thing comes to mind. Have you ever checked your spindle bearing ajustment? Might want to get the indicaters out, and just check this, make certain your spindle isnt floating around on loose bearings. Dont be surprized if they are in need of ajustment. Good luck with this.