sorry to say, but Mechanize uses LWP::UserAgent under the bonnet as far as I can gather.

Yes, thats the point, its less typing. Autocheck dies on errors, like you witnessed. No setting up a cookiejar, no saving responses... less typing is less typing :)

So it seems thast I have the latest LWP ..

That isn't exactly convincing :) LWP has many dependencies, upgrade all of them :)

You could tty  perl -MCarp::Always yourprog.pl for a stack trace to narrow down which IO::Socket... module is complaining, which module needs specific updating

Probably IO::Socket::IP and IO::Socket::SSL