If you don't want to go the route of custom certificates and an SSL-stripping proxy as suggested by Anonymous Monk, I would simply inject my own callbacks into the page. There should be examples in MozRepl::RemoteObject and/or WWW::Mechanize::Firefox::Examples somewhere how to trigger a Perl callback from Javascript, but the general gist of it should be that you don't need to think about details:

my $socket_io_handlers = $mech->repl->expr('variable_that_handles_the_
+callbacks');

$socket_io_handlers->{on_chat} = sub {
    print "socket.io: Got " . Dumper \@_;
};
[download]

This approach won't handle callback chaining in a nice way. For that, you'll have to use a proper Javascript function and inject that instead:

my $create_tee = $mech->repl->declare(<<'JS');
    function (next_handler, perl_handler) {
        next_handler.apply(arguments);
        perl_handler.apply(arguments);
    };
JS
my $old_handler = $socket_io_handler->{ on_chat };
my $new_handler = $create_tee->( $old_handler, sub { print "socket.io:
+ " . Dumper \@_ });
$socket_io_handler->{ on_chat } = $new_handler;
[download]