Measuring a very fine wire with 10 µm precision

Best option is to use an optical microscope. "Seeing" 1um is not hard at all. Get a decent CCD camera from Pixelink or Basler or any other brand with a 1" sensor and a pixel size of 2-3um. You'll also need a microscope and some basic image processing to automatically identify the wire (easy when looking for a straigth line on a uniform backgound)
It shouldn't cost more than $1000.