Animation of Tire Chains


I have to do an animation like this:

The assembling of a tire chain on a wheel. What is the best/easiest/fastest way to achieve an acceptable result. I tried rigid bodys but after baking to keyframes, the simulation is broken (I think the rotation is baked in a wrong way).
Aside from that it takes a long time to test the simulation from the beginning after every change in the animation.

In the end I need the animation of the chain with keyframes (export to Unity). I hope someone could give me a hint.