"Should someone with fibromyalgia and chronic pain do the ice buck challenge?"
Not too sure what you are asking here. Just reading about that challenge caused pain in my arms and legs. I guess it depends on what your goal is in dumping the ice water on yourself.
You should know if cold temps cause you pain or not. If they don't, then don't let extreme changes in temp stop you from doing what you want to do. If cold temps cause pain, then either dump ice water on yourself if you enjoy pain or avoid the ice water if you don't enjoy pain. Simple as that.
Or are you asking if the cold water will injure someone with FM? Not that doctors know. However, it can make them feel miserable for a while with flare-ups in many chronic symptoms.
Since FM is an invisible disease, I think we should dump an empty bucket over ourselves and then write a check for FM research!