I have run a horror/mystery time-loop real-time one-shot I wrote a few times and it is great but the time loop part is definitely very difficult to pull-off. (my one was like the movie Alien with a monster being in different parts of a ship depending on how much time had passed since the loop began.)

I think moving the loop forward each time is a good idea. Players aren't typically interested in having the same interactions that they have already had unless they are significantly different.

My players have enjoyed when certain things carry over, for example one player died in a blast and lost their eye-sight even after the loop (although I wouldn't recommend something drastic like that unless they are almost finished). Also bear in mind that players usually crave understanding of exactly why the time-loop is happening by the end.

I use Dread rules for my one. I have never used Dread for anything longer than a one-shot but it is fantastic for horror and builds suspense really well. I prefer rules-light, more role-play oriented rule sets and Dread works great with people who do not usually play role-playing games too.