Admittedly not tried it but I can’t imagine gaming via RDP would be a good experience, I guess Minecraft isn’t too graphics intensive though.
What are you wanting from the two display output? To be able to play either from the living room or bedroom or play in one location and just watch from the other?
If playing from both locations you’d probably need to look at a gaming laptop although £600 might be optimistic (although Minecraft would likely be OK). I guess a small base unit would work to – for convenience you’d want a USB dock in both locations with keyboard, mouse and local display connected so you just need to plug the dock into the laptop/base unit each time.
Personally I’d wait on a PS5, although that’s going to be a pain moving about frequently to.