I don't believe it is possible to authenticate to a proxy with just LWP::UserAgent. However in the WWW:Mechanize docs it says
If it's possible with LWP::UserAgent, then yes. WWW::Mechanize is a subclass of LWP::UserAgent, so all the wondrous magic of that class is inherited. .
So you should be able to use the same method(using HTTP::Request) to authenticate.
Update:May also want to try using the LWP::Authen modules.