I'd try what Re^5: HTTP error response code 500 using LWP::UserAgent on one site, but not on any other says

use IO::Socket::SSL qw(debug3);

Also, http://search.cpan.org/perldoc/Crypt::SSLeay#SSL_diagnostics_and_Debugging lists $ENV{HTTPS_DEBUG} = 1;

among other tips

In reply to Re: WWW::Mechanize and HTTPS proxy by Anonymous Monk
in thread WWW::Mechanize and HTTPS proxy by Gangabass

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