I've now written a shortish paper (11 pages) summarizing the main idea I had about it - i.e., to take validity as a primitive notion and add it to Peano arithmetic governed by a couple of reasonable principles:
if $\phi$ is valid, then $\phi$and a restricted necessitation rule, saying
if $\phi \rightarrow \theta$ is valid, then if $\phi$ is valid, then $\theta$ is valid
if $\phi$ is logically derivable, then infer "$\phi$ is valid"showing that one can get a consistent (indeed conservative) extension. I also add a truth predicate, along with "if $\phi$ is true, then $\phi$", "if $\phi \rightarrow \theta$ is true, then if $\phi$ is true, then $\theta$ is true", along with the principle that validities are true. This is also conservative over PA.
The basic idea of the conservativeness proof is quite simple: just replace "$\phi$ is valid" (and "$\phi$ is true") by "$\phi$ is a theorem of pure logic (in the relevant language)". Then everything comes out as a theorem of PA. Everything is classical.
The paper, "Validity as a Primitive", is here (at academia.edu) and will appear in Analysis.