WWW::Mechanize has this to say about doing SSL through proxies:

noproxy => [0|1]

Turn off the automatic call to the LWP::UserAgent env_proxy function.

This needs to be explicitly turned off if you're using Crypt::SSLeay t
+o 
access a https site via a proxy server. Note: you still need to set yo
+ur 
HTTPS_PROXY environment variable as appropriate.
[download]

Did you try that? Just so you know, the nasty hackers you're trying to outsmart - they RTFM!

-sam