I agree with hermetic, what you are measuring at the back end of the jaws is irrelevant. You mentioned adjusting the chuck body to .0005", now if that's case, adjust to the jaws, don't worry about indicating the chuck body unless it is way out of round.
I find that .003 is not acceptable.
What is acceptable to you? If you want guaranteed under .002" TIR you'll need to get a Set-Tru/Adjust-Tru/Setrite type chuck that allows you to adjust TIR.
Like when I had my HF814, I had a Bison 5" 6-jaw Set-Tru chuck. These easily cost $1000+ & the 5" ones aren't commonly stocked so no discounts on them. And that's not including the backplate which they do not make for this style spindle. I was luckily to find mine used for $600, don't forget I only paid $360 for that lathe brand new from HF. A standard back plate won't work with these chucks so I had to make my own as shown on my thread here.
I loved that chuck & I have the same Bison 6-jaw Set-Tru chuck on my current lathe but in a 6.25" size. I can easily get my TIR down to .0002" measuring on my DTI which is less than what Bison guarantees but you have to pay the price. Fuerda makes a much affordable version now, $400 for a 5"
You can also make any chuck "adjustable" by slightly undersizing the register on the spindle or oversizing on the chuck. The play will allow you to adjust TIR. This is not generally recommended though as the chuck can slip out of round. I've done this with the 3 jaw chuck that came with my current lathe, I can get it down to .0005". But I don't use that chuck much other than for polishing & light work. Grinding the jaws would be the best solution.