Ok. As per my above comment, it should be possible to filter out the analytics/sentry via DNS using pi hole. In theory, you'd just have to set up your Plex machine to use pi hole DNS ad blocking, and ensure that the pi hole domain list includes analytics domains. And Tailscale MagicDNS with the primary DNS server being your Plex/pi hole server, with your other devices configured with Tailscale, should block everything.
The only potential blocker I can see is whether the HDMI streaming device makes HTTP requests, in which case it would need to use the Tailscale proxy, and I'm not sure if that's supported.