You could do something like an LK373 transmitter (preferably a v3) - it's normally part of an "HDMI over Ethernet extender" transmitter/receiver pair, but all it really does is capture HDMI and output it to the network as an RTSP stream that you can view with ffmpeg or VLC (no need for the Receiver). They're about $45 on AliExpress.
(I use one to capture the output of a cable box at home, then proxy it through xTeVe to Plex as a live TV channel.)
Correction: model is the LKV373A
Lots of info here: https://blog.danman.eu/new-version-of-lenkeng-hdmi-over-ip-extender-lkv373a/
There are several versions of it out there, make sure to get a v3 (v2 is older and not as full-featured, v4 is partially encrypted and not well-supported).
It can be a bit tricky to get it set up initially (especially since it uses an incomplete implementation of Multicast), but once it's going it can be quite solid. Feel free to PM me if you have questions and I'll do what I can to help.