Not sure how must they cost though.
Can the measurement system touch or be affixed to it?
Sounds like a pair of nice calipers might work. So depending on your precision needs, you might get away with the same approach: sliding grid of capacitive cells that slide over the measurement cells. Microcontroller measures them as it slides through. Atan2() for the final result. The meter only part of this is called a DRO(Digital ReadOut)
If your project is not budget constrained, than there are complete closed-loop stage solutions around:
Best of luck, and prepare yourself for sticker shock... lol =3
What counts as cheap to you?
I'm thinking about automating something a long these lines:
https://youtu.be/hnHjrz_inQU?si=dNzXVBVFsr7e8m_6
Off the shelf lasers and camera sensors can be hacked around with DIY for some pretty unexpected precision.
https://www.youtube.com/watch?v=j-u3IEgcTiQ
https://www.youtube.com/watch?v=ucuVsReDze0
https://en.wikipedia.org/wiki/Michelson_interferometer
Note those 532nm green DPSS lasers are repeatable within +-1nm across their normal operating temperature. Adding a 20nm wide OD6 narrow band-pass filter to a $5 5mW DPSS laser module is the cheapest precision money can buy these days.
Really depends what one can get away with in the mechanism being built. Note, many machines will fall under export restriction, and as a company people have to decide whether that encumbrance is worth the hassle.
Best of luck =3
The term of art that I'm exploring is called "Holographic Interferometry".
Sibling poster gave you a link to regular interferometry.
But basically, if you split the laser beam, one goes straight into the camera sensor, and the other off your object, you can do some pretty amazing things. Depending on a lot of little details (The devil is hiding here).
I found 3Blue1Brown's explanation to be the best, but less "ready to use".