I don't use WWW::Mechanize, but have a look at its noproxy documentation. It says:
in reply to Re^2: IO::Socket::SSL incompatibility with WWW::Mechanize
in thread IO::Socket::SSL incompatibility with WWW::Mechanize
This needs to be explicitly turned off if you're using Crypt::SSLeay to access a https site via a proxy server. Note: you still need to set your HTTPS_PROXY environment variable as appropriate.
Even if you're not using Crypt::SSLeay, there might be something (not sure exactly what) of relevance there.