I am creating a character who's arms and legs are rigid. I have attempted to make this effect using a smooth bind but it doesn't work perfectly; it's close but not 100% correct.

Anyways, it is a setup where there are 3 joints, the shoulder, elbow and "hand" (not exactly a hand but it is where the arm ends) and the elbow bends -90° and has been limited to bend no more than that. I then painted the joint's weights by going to Edit Deformers->Paint Cluster Weights Tool and painted the weights to look like this:

I then created a blend shape for each arm where the front middle edge goes up exactly 2 units and the back middle edge goes down exactly 2 units. The blend-shape looks like this:

Afterwards I set up a driven-key where when the elbow joint bend on the X-axis is 0° the blend shape is at 0 and when the elbow joint bend on the X axis is at -90° the blend shape is at 1.000. This will give me the desired result:

I then lock the elbow joints rotate Y and Z attributes, hit insert and move is back a very small amount for the IK to work and everything is perfect... until I actually add IK which messes everything up. I can't really explain what it does so here is a video (gif):

I definitely think it has something to do with painting the cluster weights because I used the same method with the legs (only I didn't paint any cluster weights) and it worked perfect. The reason I had to paint the cluster weights on the arms is because the shoulder joint isn't at the top of the mesh (unlike the legs) because it needs to rotate from a certain point and not the top.
Can anybody help fix my problem? Thanks in advance!