+ 1 on no need for a back up camera when shooting as a second, especially when you are using a new, good quality camera such as the 6D. These (generic) machines are incredibly reliable, being designed to reliably record one off, unrepeatable events.
However as a primary photographer at such an unrepeatable event I always have a back up of everything because of Sod's Law.
For a proper back up you really need exactly the same body, as has been said, so you can share the work load and lenses seamlessly between them.
As for the 6D vs 5D III debate, well ........