I think Mark Hovey has pointed out the remark necessary to finish the proof. If we work in the injective model structure, then being fibrant is equivalent to being injective. If S are the stable equivalences then the S-local objects are necessarily injective spectra. Now use lemma 3.1.5 and a generalization of example 3.1.10 to proof that injective Omega-spectra are all S-local objects. Because we changed our cofibrations, we don't exactly get the stable model structure, but we get one with the right weak equivalences and that's what counts.
I therefore think we can conclude: there is a model structure on symmetric spectra with stable equivalences as weak equivalences, which is the left Bousfield localization of the injective model structure on symmetric spectra with respect to the stable equivalences.