I think MIME::Lite will use the default mailer (e.g. sendmail) unless you tell it otherwise. You may have more luck using SMTP as mail servers tend to treat authenticated emails with more respect. I have just asked my own question about Net::SMTP so have this snippet at hand, using Email::Stuffer. I haven't had any problems with failed deliveries.

sub send_email {
    my $msg = shift;
    my $sender = Email::Stuffer->new;
    $sender->transport('SMTP',
        host            => 'mail.example.com',
        sasl_username   => 'username',
        sasl_password   => 'password',
        helo            => 'example.com',
        ssl             => 1,
        debug           => 1,
    );
    $sender->to( $msg->{'to'} );
    $sender->from( $msg->{'from'} );
    $sender->subject( $msg->{'subject'} );
    $sender->text_body( $msg->{'body'} );
    if ( my $attachments = $msg->{'attachments'} ) {
        for my $attachment ( @$attachments ) {
            $sender->attach_file( $attachment );
        }
    }
    my $sent = $sender->send;
    return $sent;
}
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