LWP::Simple is not an object oriented module. It's not a class, so conceptually impossible to subclass. :)

You may want to subclass LWP::UserAgent instead? LWP::Simple is implemented using an object of this class.

... and if you want, you may even trick LWP::Simple into using your object instead ... just assign a properly instanciated and initialized object to $LWP::Simple::ua before requiring LWP::Simple! (Disclaimer: Probably evil.)