Wow, a very different take on the "enemies have countdowns" idea! I'll be honest though, I'm not entirely sure if there's any reason to be worried about running out of time. I always tried to wait as late as possible before killing enemies to run down my timer, and I would still always lose to not having enough damage output. Maybe I'm not strategic enough with my items?
But anyway, you should give yourself a pat on the back regardless. Making roguelikes is no joke!