I believe that what you are aiming to do is not possible with the mechanisms that you propose.
In most transcriptional and translational genetic circuits, the limiting factor for switching state is dilution or decay rate.
This is because genetic regulatory networks directly control rate of production, rather than concentration. Thus, even if your oscillator instantaneously switches on and off (which it won't), your ability to turn off an enzyme depends on the rate at which its concentration decreases.
Degradation mechanisms in the cell do not operate at the time scales that you're looking for (see, for example ,), so you need to depend on dilution. E. coli at full stretch goes at about at 20 minute division time, and even the fastest doubling organisms (e.g., V. naturiegens) max out at around 7-10 minute doubling time. That gives you a half-life, not complete removal, so if you want an enzyme to be mostly off (not just decreased some), then your target times are simply not achievable with genetic regulation.
When cells need to regulate protein activity on these tighter time scales, they generally regulate protein state rather than protein concentration. This is more difficult, but if you really need to hit such tight times, I recommend that you look into such mechanisms. Alternately, do you really need to modulate this quickly?