When is the delay? Is it from 'run the script' until 'window with button opens' (which can be attributed to Tk, but is a delay I've never seen and don't reproduce), or is it from 'click the button' until 'Browser window opens' (which is very likely something specific to your default browser and nothing to do with Perl or Tk)?

BTW, you don't need to require and use the same module - just use use.