AP,
Given all the things that you have done, my best guess is that there is something in the vehicle itself that makes it extraordinarily sensitive to any sorts of vibration. I've heard about many vehicles that have this sort of problem and fixing it involves changing the structure of the car - that is welding a brace to change the resonant frequency of chassis.
So as a first step, you should consult with other owners of your vehicle. Have any of them tried strut braces or similar stiffeners? These are the kind of things you'll find in racing catalogs. They MIGHT work, but they might not. Collectively, someone must have tried a few of these things.
Second, what was the result (in numbers) of the road force balance? Given that a tire change triggered the appearance of the problem, it just might be the tires (and/or wheels) are out of round. What you should be looking for is something under 10#. Plus, it might be the fit of the wheel to the hub.
Do you have aftermarket wheels? They typically aren't centered by the hub - which means the lugs are centering the wheels and that is prone to issues.