"If I leave the expander ball in the die, I get .004+" run-out. If I take the ball out, the run-out is .001" to .0015"."
Obviously, the problem lies with the way you are expanding, nothing to do with the sizer body itself.
Actually, the problem you have is very normal with common dies, it's due to the way we have to pull a expander back through the sized necks. All necks have a softer or thinner side that side expands more, and that distorts them. No amount of tightening or loosening or turning durning sizing will change it. To avoid bending necks we have to do something entirely different.
First choice is the Lee Collet Neck Sizer die. There is NO expander in it, just a bullet-diameter mandrel which the necks are squeezed against. Using it is different so there is a learning curve and if you aren't willing to take the time and effort to do that it is the wrong choice. But it WILL NOT distort necks as conventional expander type dies do.
Second choice is to take the expander ball out of your die, toss it and size normally. Then use a Lyman "M" expander die. It pushes IN rather than out, so the neck is well supported by the shoulder during expansion. There is much less tendency for the necks to bend toward the soft side with the M die.
With either method, it is helpful to lightly skim turn the necks a little, maybe as much as 60-80% of the circumference of the neck, just to even up the thickness.
For light neck cuts for factory rifle ammo, serious and expensive BR grade neck turners are uncalled for. If you have a case trimmer from Lyman, Forster or RCBS you can get simple attachments to turn necks from those makers.
As an excellant alternative, I'd suggest the Forster "HOT 100" hand held turner. It's inexpensive, has a carbide bit and is "micro" adjustable, few others are any better and all others cost more. Some cost a LOT more! Midway has them. Or they did.
Using these methods, I have cut the run-out of my loaded ammo to about one third of what it was before, on average. And the percentage of my cartridges with near zero run-out is much greater than before!