If the plates are not contaminated with anything (e.g., just painted or similar), you might consider an auto-collimator. They are visual devices and measure visually.
CMM has been mentioned.
Vision machine, sometimes aka visual CMM is also an option.
Either of these three should provide accuracies that you need, though you will need to study a bit to verify that.
If you have more money, then also consider a laser interferometry approach.
Hope this helps.