Don’t know of a plugin that does that but if you take a look at WP PHPList plugin or more specifically the way it Interacts with phplist using PHP CURL, it logs in to phplist as admin and save cookie using CURLOPT_COOKIEFILE and then simulates a post to subscriber form. So maybe going in that route you can do the same and simulate a post to
send mail form.
then all that is left is to hook you function to these hooks:
add_action('new_to_publish', 'send_new_post_mail');
add_action('draft_to_publish', 'send_new_post_mail');
add_action('pending_to_publish', 'send_new_post_mail');
add_action('private_to_publish', 'send_new_post_mail');
add_action('future_to_publish', 'send_new_post_mail');
I know its not a complete solution but should give you a jump start.