Yes, this approach has been tried, and we're about to submit a paper. Alas, very little is known about hyperbolic (as we call them; those with a single positive eigenvalue) graphs, and currently the proof is heavily computer aided (too many cases) and still using some algebraic geometry (each line gives rise to an elliptic pencil, and these pencils are studied arithmetically). Accidentally, just the Picard number estimate is not enough: one also has to use more subtle criteria of embeddability of a lattice to $2E_8\oplus3U$.

Segre's bound is $64$ in general (there is a gap in the proof) and $48$ in your case (no plane fully split; no gap). For the moment, see arXive:1303.1304 for further details and modern proof.

Yes, this approach has been tried, and we're about to submit a paper [edit: the paper has now been submitted, see arXiv:1601.04238]. Alas, very little is known about hyperbolic (as we call them; those with a single positive eigenvalue) graphs, and currently the proof is heavily computer aided (too many cases) and still using some algebraic geometry (each line gives rise to an elliptic pencil, and these pencils are studied arithmetically). Accidentally, just the Picard number estimate is not enough: one also has to use more subtle criteria of embeddability of a lattice to $2E_8\oplus3U$.

Segre's bound is $64$ in general (there is a gap in the proof) and $48$ in your case (no plane fully split; no gap). For the moment, see arXiv:1303.1304 for further details and modern proof.