Kenneth, I don't think the problem is the chuck if you are only seeing .001-.002" runout on the body. Anytime you remount you will most likely get runout. You can minimize this my marking the position in the jaws. An easy way to check the jaws is ti close them on a steel ruler and measure the runout. I'm not sure why you are remounting. Once I have my wood in the chuck it normally doesn't come out until I'm done and will not need to put it back in the chuck.
I have 4 chucks and I measure as stated above. All of the jaw faces were less than .002". HTC 125 (taper insert) runout was .0035, Nova G3 ( direct thread ) was .003". My Barracuda 2 and 4 both measured less than .001" runout. I've never had any problems with the Barracudas, but I usually use my HTC on my PM 90 and G3 on my midi's.
When working I had more money than time. In retirement I have more time than money. Love the time, miss the money.