Bibliographic request concerning an article by Bernstein and Robinson
(1) From the front of that issue of the PJM
I can identify one individual who scrutinized the Bernstein-Robinson manuscript and established its validity before it was published in PJM:
Paul Halmos, I Want to be a Mathematician: An Automathography (1985).
So even if the identity of Halmos as referee cannot be established, he at least did the work of a referee.
Actually, Halmos did more than the work of a referee, the preprint he received from Robinson motivated him to build on it and he wrote his own contribution, which was published in PJM (immediately following the Bernstein-Robinson paper).
As a matter of pure speculation, a reasonable guess regarding a referee could be Smith and Halmos themselves (because they posed the conjecture) or perhaps Aronszajn because he had proved the compact case.