I haven't had to do that, but maybe the following works for you?

BEGIN {
    close STDERR;
    my $logfile = '/some/hardcoded/path/where/to_put_your.log';
    open STDERR, ">", $logfile
        or die "Couldn't create logfile: $!";
};
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Put that code somewhere where it gets run as early as possible (for user code) - maybe you can even specify modules to be loaded when you start the program?