What is the best way of finding the center of an object mounted on

What is the best way of finding the center of an object mounted on a (horizontal), 4th axis?
(So you can do face milling and the top and bottom face line up perfectly. An so the diameter of objects milled using A,Y and Z is correct.)
I guess a lot of people must do that task given how common a 4th axis is.

The object is non-conductive.
I do have an edge finder.
I have an electrical tool-height-sensor to probe along any axis until the tool touches something that closes the circuit.

warning…the definition of “center” may differ from person to person. Among other things, there is the center of gravity of an object, the center of area and the center of simple outer coordinates.

The axis. The only center that makes any sense in this context.

Taking a rotation cut, okay. But how to center after that?
I an find the left and right edge with the horizontal edge-finder and thus calculate center Y.
But how do I center Z? The edge finder only works in the horizontal plane and the object is not conductive.

I have no clue what you’re trying to say with “round bit instead of cutter”.

Method 2 would work if I do

  1. cut
  2. find horizontal center
  3. move to horizontal center
  4. cut again
  5. meassure horizontal diameter
  6. move Z down by 1xradius

Method 1 would be inaccurate since I’d have to eyeball the point of contact.