Only way to balance the clutch assembly is to bolt it complete to a specialist machine. The flywheel should be balanced in its own right. The clutch plate and cover should also have been balanced during manufacture.
Vibration after fitting suggests something is damaged or misaligned.
If the gearbox weight is hung on the centreplate, it can bend the plate. (Like bending a piece of paper, it will never recover) The plate will then flex all the time it is running and will quickly disintegrate. Vibration while running can be a symptom.
If the gearbox is not properly aligned with the engine vibration is likely and rapid failure will follow. If the gearbox shaft is parallel to but not in line with the crank, the centreplate will try to run eccentrically. This will quickly tear it to pieces. If the gearbox is at an angle to the crank, the centreplate is made to flex. It will again tear itself apart, but in a different manner.
Alignment is set with dowels or dowel bolts between gearbox and engine backplate, but on some vehicles these can be displaced, or dowel holes damaged, leading to misalignment. (The Honda gearbox used on Maestro, Montego, Rover 200, 400, 600 and 800 series suffered a lot with this.) A damaged, cracked or bent engine backplate can also cause these issues. Often caused by grounding at some time.
If the vibration has only been apparent since the clutch change, this is the most likely cause. It will sadly need removing and carefully inspecting to diagnose the root cause of the problem.