The facts of the matter are not in
dispute. The pilot, so far as I know, does not deny that he basically ran the
plane dry.

There is a difference between not denying and admitting. In this case,
possibly a big difference.

No one involved seriously entertains any other scenario.

Then let me entertain one for you. There is some perfectly logical
explanation (meaning that something really unexpected happened, or
there was something the pilot didn't know about) and this situation is
a far cry from stupidly running the plane almost dry - but the pilot
has no wish to discuss it with someone he sees as not being his peer.

It's not an unusual situation. I know several airline captains - and I
can't think of a single one who would discuss such a thing with some
random member of the flying club. I don't know who the chief
instructor is, but it's entirely likely he doesn't meet with the
captain's seal of approval either. I've known quite a few club chief
instructors who got (and deserved) nothing but contempt from airline
captains.

In other words, you may be dealing with a situation that is not nearly
so cut-and-dried as you think it is, and with a pilot who believes you
have no right or standing to question him. In fact, I think this is
the most likely situation. If what he did was actually against a
specific, written club rule, you might be able to have him thrownout of
the club. If not, it might be far more difficult. You can send a
letter to his chief pilot, but unless someone can sign it with an ATP,
it will certainly be ignored.

Michael