Have you though? I’m not entirely sure that Discourse will work behind basic auth; I tried once but gave up. It seems even more likely that it might break wp-discourse (oh, and I was right about that!). If you were successful, you’d not be writing this message. 
For Discourse staging sites, I generally just set “login required” and that keeps away search engines. Someone with a login can still log in, but that’s usually what you want.
If it’s unacceptable that your sites are available to anyone then maybe you should use a firewall and limit access to your IP.
Can you tell a basic auth to allow access from the IP of your Discourse server? Perhaps: Allow access from one IP, require basic auth from all other sources - Traefik v2 - Traefik Labs Community Forum