Docker container names as client names?

That's exactly what I was thinking about. I was half-expecting such a feature to be directly available in docker.
Anyway, run this periodically and add this to a file in a separate directory you mount into your Pi-hole container. Then specify this directory using hostsdir=... in a config file inside /etc/dnsmasq.d/
New or changed files are read automatically without the need to send a signal to FTL.