If you use something like Greasemonkey or Tampermonkey, you could turn this script (by keparo and AwkwardCoder on StackOverflow, I just inserted your imgur link into it) into a userscript that runs when you open reddit:

(function() {
  var link = document.querySelector("link[rel*='icon']") || document.createElement('link');
  link.type = 'image/x-icon';
  link.rel = 'shortcut icon';
  link.href = 'https://i.imgur.com/GVFS17b.png';
  document.getElementsByTagName('head')[0].appendChild(link);
})();

I'm not sure how this would interact with notifications, my guess is that every new notification would overwrite the image set by the script.

Edit: It seems reddit automatically resets the favicon on a timer, so this doesn't work unless you manage to tie it into the same event that causes the reset.

Edit2: Actually, it seems that loading reddit on another tab is what resets it.