I would probably use ntfy.sh for this purpose. It doesn’t quite meet all your requirements, but you could use a random channel name and get some amount of security…
You can self host it, or use the hosted version. (I know it’s technically not chat, but it works on a series of messages, it just happens to call them notifications.)
These devices have been recommended in the past, and it looks like they can run OpenWRT
https://www.amazon.com/GL-iNet-GL-SFT1200-Secure-Travel-Router/dp/B09N72FMH5
https://openwrt.org/toh/gl.inet/start