Do you have any specific examples of websites that don't work? There's no reason that the current version of the plugin should be affected by whether the website uses a CMS so there might be a different problem with the site you were thinking of.
If you would like to refer to this comment somewhere else in this project, copy and paste the following link:
A Large number of websites use CMS's these days for example Joomla which stores it's favicons in:
Front-end: www.website.com/templates/CURRENT_TEMPLATE_NAME/favicon.ico
Back-end: www.website.com/administrator/templates/CURRENT_TEMPLATE_NAME(e.g. isis)/favicon.ico
If you ever wanted to add support for them? Except the template names could be anything so I don't know how you'd work that. :S
P.S. Great job so far though!
Do you have any specific examples of websites that don't work? There's no reason that the current version of the plugin should be affected by whether the website uses a CMS so there might be a different problem with the site you were thinking of.