Couldn't I just bypass that small PCB at the bridge, cut and strip the wires that connect the bridge PCB to the motherboard, cut and strip the GraphTech piezo leads, join solder and shrink wrap?
Sure, why not? Personally I'd prefer to leave the originals intact, but if you don't care about that then your approach should work fine. One thing to be careful of: GraphTech claims their wiring is Teflon insulated, and that can be nasty to strip cleanly without nicking a lot of strands or outright breaking off the piece you're trying to strip. The gold standard for stripping Teflon involves thermal tweezers, but you can do it mechanically with some difficulty. Just leave yourself some extra to allow for breakage and general "...oh, crap..." moments :-)