I cannot reproduce your problem, i.e the test script is not slow. But since your script runs without problems (except slow) even though the certificate of this site is invalid (missing chain) I guess that you are using a fairly old version of WWW::Mechanize, LWP::UserAgent and IO::Socket::SSL which does not yet check the certificates properly.

I thus recommend to first upgrade to new versions to see if the problem goes away. If not please add the full versions of the modules you've used.