Simple rig to measure current drawn in very short bursts?
An indirect way to measure the magnitude of a current pulse would be to power the circuit from a capacitor. During each burst of activity by the processor, the capacitor voltage will drop by a value that's proportional to the charge (current &times; time) consumed during the burst.

&Delta;V = &Delta;Q / C = I &times; &Delta;t / C

As long as you know what C and &Delta;t are, you can determine the value of I:

I = C &times; &Delta;V / &Delta;t

Obviously, the capacitor voltage will also "droop" over time because of internal leakage current and any quiescent current drawn by the load, so be sure to take this into account.