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I would use a fixed document camera with a height adjustable holder/desk for the actual piece of the board. The former are often fairly high quality in terms of resolution at their fixed focal length as long as you have good lightning and can often be found for cheap used. They definitely will also have lens distortion,but it should be easy to create your own profile to counteract this when doing a few calibration runs. (I think there I came around a tool for that once,but can't find it now)
The height adjustable holder/desk makes it fairly easy to always keep the actual image plane in the same distance so your profile once calibrated would fit, additionally you can use it to bring the pane of the object in the right angle to towards the camera if it isn't "flat" enough. I personally would try to adopt a 3D milling table from AliExpress for it, put it in a box below the actual image pane and use a digital angle meter to make sure it's actually flat
With that you should be able to achieve an accuracy below 0.1mm at least