Would you be able to post links to the devices you’re looking at to get some context?
If these devices are what I’m thinking of then they don’t typically run through a switch/router. Instead, they’re meant to run directly connected from device to device as if they were one long cable. The biggest downside to these types of devices is the latency and bandwidth limitations they introduce. Some of these HDMI over Ethernet devices cap out around 1080p60 (maybe even as low as 30 hz) and can add noticeable latency. With that said, I don’t think the usb over Ethernet should introduce a noticeable delay and think that one is fine — if it’s what I’m thinking of.
Have you considered using a fiber optic HDMI cable? I have had success running a 50’ fiber optic DP1.4 at 5120x1440p120 with no noticeable latency (though I don’t play a lot of FPS) that I bought off Amazon for like $80. If you could stretch your budget a little more perhaps the usb over Ethernet (or fiber) paired with a fiber cable for HDMI might suit your needs.
Alternatively, if you aren’t gaming at super high refresh rates and latency isn’t super critical, then streaming to a device connected to the tv in your living room might be a better choice. I’ve run both a raspberry pi 3/4 and an nvidia shield using steam link and found that to be great when gaming from different rooms in my house. Using a wired connection end-to-end the network latency is usually within the frame-time when playing at 60 fps and hasn’t been noticeable for me (again I don’t play a lot of FPS games).