Measuring low-duty-cycle supply current
This is a difficult problem. What I have done in the past is measure the current during the brief high current event using a current probe on an oscilloscope. In my case, the current profile was the same every pulse. So then I integrated to figure out how many coulombs each pulse is. Then I could just count the pulses. In my case, the current consumption in between pulses did not make much difference to battery life (which is what I was trying to figure out), so I could just ignore it. But you should calculate it to be sure. An ammeter should be fine for measuring current between pulses.

If your current pulses are variable in length or amplitude, this approach may not work very well.

Another idea is to use a rechargeable battery of known capacity and measure time until the battery reaches the termination threshold (you can log VBAT with a logging voltmeter). This is probably what you are trying to figure out anyway, right, battery life? In any event, if you use a small battery, that could allow you to get a reasonably good answer in a reasonable time frame, as long as the battery can supply the pulse current without excessive drop.

Things to consider are the variation of current with voltage (in my case it was constant current, but if you have DC-DC converters it may be more like constant power).

Good luck.