The standard high-precision way to use a three-plate differential capacitor movement transducer is a transformer ratio arm bridge, which was invented in 1929 (British patent 323,037) by Alan Dower Blumlein. The subsequent US patent is 2,564,390. The advantage of Blumlein's approach is that stray capacitances are prevented from affecting the bridge balance.
Here are some discussions:
"Introduction to Instrumentation and Control" By A. K. Ghosh, Published by PHI Learning Pvt. Ltd., 2004, ISBN
8120316266, 9788120316263, 344 pagesPage 93. Transducers and Instrumentation By D. V. S. Murty Published by PHI Learning Pvt. Ltd., 2004 ISBN 8120309537, 9788120309531
572 pagesbad scan. described advantage of transformer ratio arm bridg for handling stray capacitance Comprehensive Transducers for Instrumentation By M.G. Joshi Published by Firewall Media, 2005 ISBN 8170082161, 9788170082163
Alan Blumlein also invented stereo, and a host of other things.
Joe Gwinn