Anything in contact with either side of the piezo is going to affect frequency response. You will always get the best response when the piezo is tightly coupled to the instrument. This is why our “start here” design calls for the piezo wrapped in one smooth layer of electrical tape, one thin layer of copper, and then bonded tightly to the sound board with the provided piece of very-sticky double-tape. Beyond that, everything is an experiment. When using clamping pressure from above, instead of adhesive from below, everything counts. We think you will generally achieve best coupling when the clamp pressure is distributed uniformly across the back of the disc. We once used a piece of thin, dense foam, then a disc of 032 aluminum between the disc and the clamp.
The board-under-the-bridge idea sounds great! Would this “paddle” extend over a sound hole, or off the side of a narrow-bodied instrument? Obviously, the material will strongly affect the tone. Ideally, the paddle should be the same material as the sound board. That’s easy if you are making your own instruments. Otherwise, I would recommend 1/16″ thick basswood, such as from Midwest Wood Products, distributed through Utrecht Art Supplies and many hobby shops:
Basswood from Midwest at Utrecht
This cuts with a very sharp X=Acto knife or a micro-coping saw. Experiment with shapes and sizes. I think the length of freespan tab (or tongue) would influence resonant frequency response. Let us know what you find out!