Isomorphism preserving transformation CNF to Graph?
I think there's a straightforward transformation to graphs with colored edges, which can in turn be transformed into ordinary graphs.
Given a CNF $\phi$ with clauses $c_i$ and variables $v_i$, construct a graph with vertices $c_i,v_i,\neg v_i$.  Add black edges between each clause $c_i$ and each literal in it.  Add a red edge between each variable $v_i$ and its complement $\neg v_i$.  This transformation maps isomorphic CNFs to isomorphic graphs, and vice versa.
(Proof: Given $\pi,\pi'$, you obtain a mapping on vertices: map clause $c_i$ to $\pi'(c_i)$ and map variable $v_i$ to $\pi(v_i)$ and $\neg v_i$ to $\neg \pi(v_i)$.  You can verify that this respects the edges.  Likewise, you can convert any mapping between the two graphs into $\pi,\pi'$.)
There are standard reductions that allow you to reduce colored graph isomorphism to graph isomorphism.  You basically use a gadget to represent the colors (you attach each red edge to a copy of a gadget that is unique to the color red, and attach each black edge to a copy of a gadget that is unique to the color black).  If you compose that with the construction I outline above, then you should get the desired reduction from CNFs to graphs.