I'm not sure I'd guarantee 5 microns, but with experience the size and repeatability can be pretty good. When I was the jig borer operator on my shift many years ago we used telescoping gauges. It's dependent on feel and experience.
The previous operator who taught me said the process was: Insert the gauge a little canted in the hole, snug up the lock but not too much, wiggle sideways to be sure you're across the largest section of the bore and then draw it through center by tilting JUST ONCE. Take it out and measure with the mic, again waving back and forth between the anvils to be sure you cross center while you slowly close the mic being sensitive to exactly the same resistance through the anvils that you felt in the bore.
One repeat job was grinding steel rings that would be shrunk fit onto carbide blanks. A little tolerance but not much between cracking the rings and having the blanks just fall out.
I expect the key to doing it well is to be able to spend some time practicing with a known workpiece so you get the feel and consistency. Even if you find your measured size is different from the gauge ring, if you can be consistent you know what to calculate for the actual size.
I was told that the guy who followed me on that machine when I moved got the shrink fit job and when everything cooled the carbide blanks just fell out the bottom. It's all in the "feel." Can you sell that to your foreman (or to the women)?