$ua->ssl_opts(verify_hostname => 0);

See LWP::UserAgent and search for $ua->ssl_opts.
Update: Gah. The OP was talking about WWW::Mechanize.

WWW::Mechanize passes its constructor arguments up to its parent (LWP::UserAgent). Try this:
use WWW::Mechanize;

my $mech = WWW::Mechanize->new(
    ssl_opts => {
        verify_hostname => 0,
    },
);

[download]

In reply to Re: Mechanize and Bypassing SSL Certificate Check by repellent
in thread Mechanize and Bypassing SSL Certificate Check by Anonymous Monk

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