Given the state of material science, I'm afraid you're out of luck.
Clearly, you envision the payload energy being stored in a capacitor which is charged by the rail voltage, but you haven't looked closely at just how much energy is required. The PopSci article suggests an output voltage of 500 volts at 1.3 mA for 20 seconds. The energy required is then $$E = V i t = 500
\times .0013 \times 20 = 13 \text{ Joules}$$ Assuming the capacitor is charged to 500 volts during launch, $$E =\frac{CV^2}{2} = 13$$ and $$C = \frac{2\times E}{V^2} = \frac{26}{250000} = ~100 \text{uF}$$ and since this would produce a capacitor which is completely discharged (no voltage) at the end of the 20 seconds, a factor of 10 should be applied, for a final value of 1000 uF at 500 volts. Such a capacitor does in fact exist, but it's much larger (40 mm dia x 105 mm long) than the taser round (18.5 mm dia x ~75 mm long). Worse, the capacitor alone weighs 170 g compared to a total taser projectile weight of 25 grams. Figuring a final projectile weight in the vicinity of 250 g, a one-handed launcher (pistol) would probably break your wrist.