Although frequently labelled as one, I'm not at all the sort of hardline proof-theorist who finds no room in his philosophy of logic for model-theoretic techniques. (Does the hardliner exist? Yes - check out this paper by Tennant.) Quite the contrary, I believe the two approaches complement each other, both being equally valuable in formal methods broadly construed. Nevertheless, for particular purposes in one's philosophy of logic, preferring one over the other is sometimes right. Lots of philosophers share the suspicion that inference rules are somehow integral to acquiring and possessing logical concepts. And, accordingly, a lot of energy has been put into exploring the link between this idea and entitlement to infer. (Examples are Christopher Peacocke, Paul Boghossian, and Crispin Wright.)
Correspondingly, proof-theory has its own psychological counterpart. In fact, the mental model theory is a response to the idea that inference is a result of a rule-based mental capacity which relates to formal calculi. Byrne calls it the formal rule theory. As its model-theoretic rival, the formal rule theory offers experimental support for why reasoning as a psychological process is a largely rule-based affair. One source for this view is The Psychology of Proof (1994) by L. J. Rips.
You don't have to subscribe to experimental philosophy in its most rampant forms in order to see the value of connecting the above experimental work with the philosophy of logic. In fact, my hunch is that philosophers - although experimentally crippled (aka "don't try this at home") - can contribute to the experimental work with logical expertise. Actually, philosophers of logic have hypotheses about actual human reasoning that are primed for testing, and which aren't likely to be tested by psychologists who don't know discussions about, say, vagueness and semantic paradoxes. Earlier I've mentioned one admirable adventure into experimental research by Dave Ripley, and another by Jeff Pelletier.
My plan is to start out by checking out The Rational Imagination (2005, MIT Press) by Ruth Byrne.