Simple computing, algebraically or by software, will neglect the local instabilities. Though, runaway must start and worsen locally.
The very beginning of the reaction, when the change in concentrations can be neglected, must still be accessible to hand computation. But later, the equation mixes sums and exp(-1/T), I wouldn't even try.
I suggest to misuse Spice to conduct numerical computations. It is an extremely stable simulation program meant for electrical engineering. I find it little useful for EE, but here it would solve your query instantly, and with your narrow set of time constants, be very accurate.
You just have to represent your reaction as an electrical circuit. For instance, the amounts of reactants and products could be voltages or charges across capacitors. You specify the initial amounts. The rate of reaction would then be a current in a current source controlled by a function you choose arbitrarily, which would be the activation energy thingy. The temperature could be a charge in an additional capacitor.
There may even be some built-in activation functions, because leakage currents in diodes often follow them. Then the leakage of a diode would represent the reaction rate. Just find a way to vary the temperature over the reaction time.
Spice is free and runs on many OS. It takes one day to learn. It will solve one case instantly, so you can make many runs and vary the parameters.