I don't use Mechanize much, but I was under the impression you could do

#untested
use LWP::UserAgent;
use WWW::Mechanize;

my $agent = WWW::Mechanize->new();
my $ua = LWP::UserAgent->new;
$ua->protocols_allowed(['https']);

$agent->get('https://xxxxxxxxxxx.com/file');
[download]

also see Downloading large Binary files from https protocol using WWW::Mechanize in Windows OS