robtillaart:
The place of the wire will change the resistance of the V as the level where the wire is short-cuts the current.
I don't think that would work, because the wire is insulated. Using a camera and microscope seems like the most likely approach to me, but the other option would be to create a rolling version of a standard wire gauge (using a pair of opposed precision ground rollers with a slight taper, and angled so that they tended to pull the wire away from the narrow end of the V as it rolled between them). The height of the wire in the gauge would tell you the thickness of the wire to whatever precision you wanted, as long as your rollers were sufficiently accurate.