Sharp threads with barrel probe?

The engineer where I work wants us to measure sharp threads in a part with a barrel probe to find true position on the minor of the threaded hole. I already explained to him that in order for that to work accurately the part and the threaded hole would have to be EXACTLY lined up correctly to the Z-axis every single time and suggested that they bring us the part with the threaded gage in it to touch off of.

Does anyone have a better idea for the measurement of sharp threads? Is there a stylus type that I am unaware of that would make this possible at all?

Thanks and hope everyone is well here!

Tami