As an alternative to Expect, you could also use IO::Pty directly, as you're not really interacting with your bsub program. IO::Pty is the pseudo terminal that Expect is using under the hood.  In its most simple case this could look something like this

use IO::Pty;

sub run_in_pty {
    my @cmd = @_;

    my $pty = IO::Pty->new();
    
    my $pid = fork;  die "Couldn' fork: $!" unless defined $pid;
    if ($pid) {  # parent
        $pty->close_slave();
    
    } else {     # child
        my $slv = $pty->slave() or die "Couldn't get slave: $!";
        close $pty;
    
        open STDOUT, ">&".$slv->fileno() or die "Couldn't reopen STDOU
+T: $!";
        open STDERR, ">&".$slv->fileno() or die "Couldn't reopen STDER
+R: $!";
    
        exec @cmd;
        die "Couldn't exec(@cmd): $!";
    }

    return join '', (<$pty>);
}

my $cmd = "./bsub";

my $out = run_in_pty($cmd);

print "$cmd: '$out'\n";
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Now, with this bsub-replacement (for demo purposes):

#!/usr/bin/perl

if (-t STDERR) {  # connected to terminal?
    print STDERR "msg-to-stderr\n";
}
print "msg-to-stdout\n";
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you should get both messages with the above run_in_pty routine, while if you'd do on the commandline:

$ ./bsub >output 2>&1
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you'd only have "msg-to-stdout" in the output file.

I'm afraid this doesn't work on native Windows, though... (Cygwin should be ok),  so hopefully, someone else knows a solution for this platform.