Your second sentence is correct. So yes, the structure is possible. In this context it is awkward, stilted.
"Rests with" is an expression I would use only when there is a really good reason. For example, in newspaper headlines it is a darn useful little phrase to express a concept in minimal words. For example, "Gingrich fate rests with Iowa voters."
It can also be poetic, overly formal--which is great if that's what you want. "My future rests with you. Will you help me?"
If you want to make the first sentence clear and proper English I would say "The task of dismantling.... rests with the crew."