I'm not sure if this will work within your larger setup, but consider duplicating the Valve bone to a "hand rotation" HandRot bone, parenting HandRot to Valve, and changing the child-of constraint on the hands to the HandRot parent:

Now, you can animate the turning of the valve with the Valve bone followed by a counter-rotation of HandRot to bring the hands back to resting position. Here's a 100 degree rotation of Valve followed by a -100 degree counter-rotation of HandRot:

You can then further animate the hands to make the return to resting position look natural (which may or may involve adjusting the influence of the child-of constraint).

Additional rotations require continued and cumulative rotation of the valve bone and counter-rotation of the hand rotation bone, but the hand key frames will be local to the hand rotation bone (as long as the child-of constraint is active), and can be copied and pasted for each turn.
Here's the final animation -- I upped the rotation to 120 per turn, keyframed three cumulative rotations and counter-rotations (so the the final rotations were 360 and -360 on the Valve and HandRot bones respectively) across 60 frames, and created a 20-frame Hand.L/R animation that is copy-pasted twice, for a fully cyclic animation:

Here's my Blend file with this example.

Note that if you've already fully animated one cycle of the hands relative to the Valve bone, this technique will still work fine. Hold off on the counter-rotation of the HandRot until the hand animation has completed its return to rest position. Because HandRot has the same rotation as Valve, the existing animation will work fine. After the final frame at rest position, add a "Constant" interpolation frame to apply the full counter-rotation of the HandRot bone and copy and paste the first hand key frame to return the hands to rest position. Visually, the final rest position frame and this constant interpolation frame should look identical, but the latter will have the hands in the same local position as the first frame of the cycle.