The thinner wire has a history that it did not improve the hysteresis (ask Jenne). Nevertheless, it's worth to try.
If you flip the clamp upside-down, you can lift the clamping point up. This will make the gravity restoring torque stronger.
(i.e. Equivalent effect to increasing the mass)
Luckily (or unluckily) the clamp has no defined location for the wire as we have no wire fixture.
Therefore the clamp will grab the wire firmly even without milling.