i think perhaps a spline IK might help you here.
Try creating a high res joint chain that corresponds with the links in the chain and shape the spline curve as you want the chain path to follow.
When you move the root of the joint chain, the joints will follow the spline path.
You will also want to look into a stable way to control the twisting of the joint chain too - for that I'd recommend a system similar to the ribbon spine.