It’s easiest to just make the weight the same on both sides but to get it exact you will probably need more weight and/or different dampening on one side than the other.
The clutch side has the grip attached solidly to the bar and the throttle side has that loose fitting grip so is not as coupled to the bar as the clutch side.
Back when I used to Work for a living I would sometimes use stick on wheel weights to effect the vibration of automotive chassis parts. Those stick on weights make it super easy to change the placement and/or mass. Once the weight location and mass was understood it was then an easy task to add internal weight or build a simple tuned dampener to lower the amplitude.
It also helps to identify the antinode, if you can improve it there you are usually golden at the ends.