Bad news: After the reset, I had a longer period without issues, but after nearly a week the disconnection happened again

— sorry for the false hope!
Took the unit straight to the Apple Store where they replaced it (both TV + remote) under warranty. Now just playing the waiting game to see if the new one holds up...
To all of you getting replacement remotes for this issue: I think you're wasting your time. I really think this is an Apple TV bluetooth problem (either software bug or hardware). I was also having intermittent problems connecting my Apple Watch to use with Fitness+ workouts (ALSO resolved immediately after an Apple TV reboot) which reinforces this. I pushed back on the idea of just replacing the remote for that reason.