The 66309/66311 cannot stream data back to the computer. I believe that the later models added streaming capability, so the record could be continuous. Also, the 66309D does instantaneous sampling (instead of integrating the current over a long period). The sampling period can be set as low as 15us with 4096 samples in a row (about 63ms total). Then, you can acquire the mean if those samples without downloading each sample. The sample rate can be nearly continuously varied. You would lose any current consumption data between acquisitions, but unless there are out-of-family pulses happening I those times, it'd be ok (just assume the current remained constant during the missing periods).
You probably could add a capacitor (check documentation for stability concerns first) to smooth out the pulses to make them low enough frequency to sample.
I started to write some control software for the power supply (Microsoft Windows based), but it's pretty buggy and doesn't have the features you need, but could potentially be helpful for you.