You can indeed play games with references closures. Here is a solution that does both:

package MyModule;
use File::Rsync;

# In some local scope
my $rsync = File::Rsync->new({
  outfun => bind_args(\&capture_messages, \ my $out),
  errfun => bind_args(\&capture_messages, \ my $err),
});

# Somewhere else
sub capture_messages {
  my ($string_ref, $message, $type) = @_;
  $$string_ref .= $message . "\n";
}

sub bind_args {
  my ($func, @args) = @_;
  sub {$func->(@args, @_)};
}
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A bit sophisticated, but now the output is captured in a variable that is local to wherever your rsync declaration is done.