This is actually a tricky situation. Adding such constraints leads to a cyclic dependency, here is what the Blender console is printing:
Dependency cycle detected:
OBrig/arm_ik.r/BONE_DONE() depends on
OBrig/arm_ik.r/BONE_READY() via ‘Ready → Done’
OBrig/arm_ik.r/BONE_CONSTRAINTS() via ‘Constraints → Ready’
OBrig/arm_ik_nostr.r/BONE_DONE() via ‘Copy Transforms’
OBrig/arm_ik_nostr.r/POSE_IK_SOLVER() via ‘IK Chain Result’
OBrig/c_hand_ik.r/BONE_DONE() via ‘IK’
OBrig/c_hand_ik.r/BONE_READY() via ‘Ready → Done’
OBrig/c_hand_ik.r/BONE_CONSTRAINTS() via ‘Constraints → Ready’
OBrig/arm.l/BONE_DONE() via ‘Child Of’
OBrig/arm.l/BONE_READY() via ‘Ready → Done’
OBrig/arm.l/BONE_CONSTRAINTS() via ‘Constraints → Ready’
OBrig/arm_ik.l/BONE_DONE() via ‘Copy Rotation’
OBrig/arm_ik.l/POSE_IK_SOLVER() via ‘IK Chain Result’
OBrig/c_hand_ik.l/BONE_DONE() via ‘IK’
OBrig/c_hand_ik.l/BONE_READY() via ‘Ready → Done’
OBrig/c_hand_ik.l/BONE_CONSTRAINTS() via ‘Constraints → Ready’
OBrig/arm.r/BONE_DONE() via ‘Child Of’
OBrig/arm.r/BONE_READY() via ‘Ready → Done’
OBrig/arm.r/BONE_CONSTRAINTS() via ‘Constraints → Ready’
OBrig/arm_ik.r/BONE_DONE() via ‘Copy Rotation’
Detected 4 dependency cycles
It’s a little farfetched, but this means that some bones depends on each other, and thus cant’ be evaluated properly:
c_hand_ik.l depends on arm.r, while arm.r depends on c_hand_ik.r, while c_hand_ik.r depends on arm.l, while arm.l depends on c_hand_ik.l
You can see the loop issue, c_hand_ik.l depends on itself**.**
Then such a setup is unfortunately not possible, it leads to issues.
Instead I would recommend to parent both c_hand_ik.l/.r to the same object/bone, out of the arm hierarchy, to solve this problem.
So in conclusion, you will just have to set this up differently.