Various solutions are possible depending on your exact requirements.
This is probably the most obvious system, consisting of 1 or 2 lever arms which are deflected by the plate. By measuring the deflection the size could be determined. The main drawback of this is that it is a contacting system which may be a problem depending on your setup.
Another alternative which would be simple to implement is a video system. Mount a camera above your sample and image the plate as it passes through. By either calibrating the camera before hand or mounting a line scale beneath the plate the size can be determined. Resolution will depend on resolution of the camera and scale, although I think 1 mm shouldn't be hard to achieve. This is probably what I would do if you didn't want a contact system.
These may be possible depending on what you consider low cost. Most systems are based on time of flight, which limits the resolution to ~1 mm. Additionally most cheap systems I have seen have a minimum range of at least 5 cm, which may be too large for you.
Also may be possible. However, simple fringe counting methods have difficulties when measuring steps, so will probably not be suitable for measuring a plate (unless its infinite). Frequency scanning methods are possible, but probably don't fall under low cost, measurement speed may also be an issue depending on sample length and throughput required.