Retrodeck in your case then if you just want a simple setup.

There’s a docker image called wg-easy I use with a nice web interface.

Based on the hardware you have, if you’re able to install it on one of the routers, I know that OpenWRT can set DNS on a per device basis with it’s DHCP server.

My personal suggestion is to use a PFSense device as there is an option for per device DNS settings, along with a lot of other options that you might want to mess with if your trying to learn some networking

Armbian with CasaOS

I can’t speak on moonlight specifically on the pi 4, but it running steamlink on wasn’t bad on it, so I can’t imagine it’s would be any worse than that for moonlight.

Wired Nvidia shield TV has been my best experience when it comes to both moonlight and steam link out of all of the devices I’ve tried it with.