As long as you are able to generate a joint terminal distribution, any model will do the job. Copula is only one such approach.
Now, in theory, you cannot completely statically replicate this payoff in general. To see this, know that all you have is vanillas, and the most you can do is imply the marginal distribution from them (the usual risk neutral density). However, your exotic is also sensitive to the conditional distribution, which the vanillas cannot capture.
Even if you do assume that you have spread options (like you say in the comment), it is still not enough to determine the joint distribution completely; as you can only determine the distribution of their difference (so you now have 3 marginals). To see why, note that you cannot hope to find the moment generating function of the joint at many points (say [1,5] for instance). So you can't pin down the MGF (equivalently, the joint distribution).
One can say something stronger: You cannot find the generating function for any random vector f except the spread itself.