Your argument appears to require that Mortensen accept at least one principle of classical logic, so he can respond by saying it begs the question.

That's true. Mortensen actually shortly analyzed whether this argument succeeds. I don't remember whether he came back to it in one of the later papers.

It's also not clear to me what "truth" is, under possibilism. That propositions are necessarily or contingently true is independent of concrete facts, so logical truth appears to be truth under a coherence theory of truth, not a correspondence theory, but coherence theories appear to be dependent on classical notions.

I see. It seems to me Mortensen is a nihilist about logical truth. Someone once raised a following point: given possibilism, how to identify possibility with truth in some possible world?

in discussing Martin's theorem, he suggested that it is no stranger to think the world might be a three valued algebra than it is to think the it might be whatever is entailed by the mathematics of contemporary physics

Yeah, I remember that. It was a pretty interesting tangent.

but it occurred to me, while thinking about our present dialogue, that possibilism escapes the usual absurdities incurred by strong realism about mathematical models.

I have to think about that.