I do see another warning now, "Peer certificate not verified".

That's presumably because you have have left verify_hostname as zero. Test:

use strict;
use warnings;

use Test::More tests => 2;
use LWP::UserAgent;

my $ua = LWP::UserAgent->new;
isnt get_cw ($ua), 'Peer certificate not verified', 'Verify = no warni
+ng';
$ua->ssl_opts (verify_hostname => 0);
is get_cw ($ua), 'Peer certificate not verified', 'No Verify = warning
+';

sub get_cw {
    my $lpw = shift;
    my $res = $lpw->head ('https://www.perlmonks.org/');
    my $cw  = $res->header ('Client-SSL-Warning') // '';
    return $cw;
}
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