we can probably get away with overdriving the LED quite a bit because of it's intermittent use.
Probably not, actually: many modern LED chemistries are not as tolerant of overdriving as the ones of 30 years ago. Back then, you could usually do peaks of 5, maybe even 10, times the typical 20mA, as long as the average stayed low.
But I grabbed a Luxeon datasheet, and they recommend a peak pulsed current of no more than 500mA.
There's a really wide range of Vf for those LEDs: 2.8-4V. If your 5V supply is spot on, and your LED is near the high end of that range, you don't need a resistor at all: the voltage drop across the TIP120 is enough to keep the current low. Otoh, if the +5 is near the upper end of the regulator's 5% or 10% tolerance, and the LED is from a batch near the 2.8V end of its range, having no resistor could mean "instant release of the magic smoke from the LED".
If you don't have the tools (a really accurate multimeter and a current-limiting power supply) to precisely measure the Vf of your LED and the output of your power supply, the 3.3 Ohm resistor should be a pretty safe bet: it'll drop about 1.1V at 350mA, which will protect the LED even if all parts are near the worst-case ends of their tolerances. If you know your Vf is higher and +5 is lower, a smaller resistor would be okay, but may not make a dramatic visible difference because the human eye response is not linear.