It's been a while since I've tried to mech from inside a proxy. (Yes, I've just verbized the short name for WWW::Mechanize).
Give that a shot, or try following the other suggestions frm the FAQ, for which you've already got the link above.
If that fails (and I don't see how, since now you've confirmed you're behind a proxy) there are some other troubleshooting steps you can try near the end of the Mech FAQ, including checking $mech->status after every attempt to get something.
s**lil*; $*=join'',sort split q**; s;.*;grr; &&s+(.(.)).+$2$1+; $; =
qq-$_-;s,.*,ahc,;$,.=chop for split q,,,reverse;print for($,,$;,$*,$/)