Ehm… 
I forgot the Nginx (Discourse) part. Thank you! 
I just edited app.yml and ran a rebuild on the container. The bot came back almost instantly after the container came back up. I do not see any 429 errors, yet. Hopefully it stays that way for “normal” users.
That’s the thing… When I check the admin page of the users, I always saw different “last IP” entries. So somehow Discourse saw the real IPs of the users, even without mod_remoteip and also without the Nginx config change. 
Anyway, I am curious to see if the Nginx config change brought the solution to this problem!