I think with is the only acceptable choice.
The definition because of is prefectly correct.
because of all this work, we cannot go
But with signifies something else.
1 having or possessing:
With a bit of luck, we should be back in time for dinner."
Cambridge Dictionaries Online - Cambridge University Press
So it can be taken to mean having or possessing in a very general sense.
With all the money in the world, I'd still enjoy camping.
With your lazy attitude you are bound to be unhappy.
Back to your sentence:
With all this work (all this work that we have to do), there's no way we can go out tonight.
Student or Learner