The game it most reminds me of is Betrayal at House on the Hill. You draw room tiles as you explore, and when a Haunt is triggered - drawing an Omen card, which forces a roll, and if too low, the Omen begins, in a form dictated by the card drawn and the room it was drawn in - is (usually) one player vs. the rest with specific victory conditions of each. I haven't played a scenario exactly what you're describing but there may be one (there are fifty different scenarios and I've only played it a half dozen times or so).

That being said, the flavor of Don't Breathe is a sci-fi space station, not an old haunted house; I love the idea of scent markers and blood trails. Definitely has potential and I'm very interested in playing. If you haven't yet, check out Betrayal and look at how it sets the various Haunt victory conditions. It may give you some ideas.

Incidentally, I came up with the idea that if the monster enters your room, you would have to hide by holding your breath until rolling a difficult number (via a pair of d6). If you fail, the creature finds you.
Alternately, if holding your breath is dangerous for some reason, you could just use a short timer

Hence the title.