Depending on your version of IO::SSLeay (or whatever), LWP doesn't have much bearing on the proxy to be used for HTTPS. In one situation, the following worked:

BEGIN {
    $ENV{HTTPS_PROXY} = 'http://myproxy.internal.corion:8080';
    $ENV{HTTPS_PROXY_USERNAME} = 'Corion';
    $ENV{HTTPS_PROXY_PASSWORD} = 'Corion';
};
use Crypt::SSLeay;
use WWW::Mechanize;
...
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