I've got the skeleton of a companion container for nginx-proxy/nginx-proxy that will use RFC2136 to add/remove CNAME entries for proxied containers as they are created/removed.
I'll push it to GitHub from my Forgejo instance once it's a little bit more polished
This is in a similar vein to my one that adds mdns/avahi CNAME to the host machine.
https://github.com/hardillb/nginx-proxy-avahi-helper