I once had a go at the wincharger type fly governor and had similar problems.
For a simple fixed pitch wooden prop static balance is good enough and as long as the blades are similar the aerodynamic forces seem to be no problem.
When you have heavy metal parts moving about, things are not so easy, they really need to be dynamically balanced. Even if you make them identical as far as possible it seems difficult to get everything to move exactly together and stay in balance.
I agree that the first step is to lock the governor and be sure all the blade pitches are the same and then check for balance.
If all is well try locking with the blades at a different angle, if still ok it would indicate that the blade pitches are not tracking when governing.
Make very sure that the control weights are identical in weight, radius and angle as close as you can get them, they are the most likely things to cause a couple when they move as they are the heaviest bits to have an effect.
If you are really in trouble it may be worth removing the blades and checking the hub for balance at various speeds by driving it from a motor and belt or something.
Flux