Untested

# refactor each $router to have a 'success' key of 0;
# running this way allows you to iterate through the 
# entire list of routers, dealing with failures only
# after the entire stack has been processed.  
# assuming a sufficient number of targets, this should
# sidestep your problem as I understand it.  
for (my $i = 0; $i++; $i < 3) {
  my $failures = 0; 
  foreach my $router (@routers) {
    if ($router->{'success'} != 1) {
      my $output = `expect -f show_running_config.exp $router->[$IP]`;
      if (write_config($output, "$router->[$NAME].cfg") == 0) {
        $router->{'success'} = 1;
        $router->{'msg'} = "Output was $output";
      } else {
        $failures++;
        $router->{'msg'} = "Failed to get config for $router->[$NAME] 
+at $router->[$IP]: $output\n";
        if ($i == 2) {
          print $router->{'msg'};
        }
      }
    }
  }
  if ($failures < 1) { last; }
  sleep 5;
  $i++;
}
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-- Edit: removed unneeded $failures. -- Edit2: Thanks Laurent_R for proofreading.