Help needed trying to build an audio sawtooth oscillator
You need no fet. You can add another resistor to pin 2 of U1. It makes the integrator summing. If the new resistor is small, the time constant will be low. Draw that way the capacitor empty fast. In practice you can connect a diode from U2's output to 2/U1, anode to 2/U1. The diode prevents the loading of the cap.

It generally is not a good idea to have high pulse current reset. It will not be fast enough for linear Volt/Hz dependence. You will do better by having a triangle oscillator and a switching circuit that inverts the other ramp of the triangle voltage to another qrowing sawtooth.