I think this is one of the things that the Apollo fork[1] makes way easier. I'm not sure that I had to do anything Apollo config wise* . I just went through the Windows HDR calibration tool the first time to tune it for my TV and it was good after.

*I do have a once-per-run Moonlight glitch that may even be because I'm on an older version. The first connection just doesn't pick up on the HDR immediately. I just have to reconnect once and it's good after that.

[1]: https://github.com/ClassicOldSong/Apollo ("Built-in Virtual Display with HDR support that matches the resolution/framerate config of your client automatically")

They just updated the moonlight client which I’m sure you’re on should be more reliable and I highly recommend https://github.com/Nonary/Vibepollo fork of Apollo