I don’t quite see the problem. If there is a center punch mark on the piece, I run a sharp live or dead center (in the tailstock) up against the part, holding it against the chuck, then adjust the jaws as needed. I suppose I could use a wiggler with the needle tip to verify centered...maybe use an indicator against the needle tip. I would wonder how accurate the long pointer really is?
In fact, maybe another thread is sources of error that are not that obvious...sorry to go off topic.