You shouldn’t have to do anything to get xmlrpc.php to work. It sits and waits for incoming requests. You might be getting the 406 error because browsing to that file isn’t how it’s meant to be invoked.
I recently completed integration to wordpress from FotoFlix and it worked great. I haven’t tried posting from Flickr but I was able to post from FotoFlix without any problems.
Are you entering your endpoint (https://www.yoursite.com/wordpress/xmlrpc.php)?
If you continue to have problems I can try to post a photo from FotoFlix and see if there’s something weird with the xmlrpc.php file you have.