Will this approach accurately measure capacitance and infer spacing?
You might consider a geometry that varies the OVERLAP of the plates instead of distance. Your capacitance will vary linearly with the overlap. C varies as 1/d, so as it stands, your sensitivity at the far point will be cruddy.  Even changing to overlap, I wouldn't count on 1% accuracy.  

Consider the other options already mentioned, or an LVDT.

UPDATE:
As a followup, many measurements such as this are improved by a push-pull arrangement.  If you can work this out using TWO capacitors, where one gets bigger at the same time and rate as the other gets smaller, sensitivity and linearity will both improve.