Thanks for this comment. I had seen that paper, however we haven't been able to adapt their techniques for this more restricted case. The switching allows one (along with local complementation) to span all possible graphs on v vertices. The proof in that paper relies heavily on this fact. For LC alone however one is limited to subsets of all the graphs, and as far as I known there is no efficient method of calculating the number of graphs in an LC-orbit. I suspect this is part of the reason why this problem may be difficult to answer.

Stated in this context, if I'm not mistaken, the question becomes: (if true!)Why is the local complementation group on k graphs (in an lc-orbit), either k-transitive or (k-2)-transitive?