0.4A for 4.7ms will cause a 10,000 uF capacitor to droop by about 190mV. If a pulse like that happens once an hour it is an average of about 0.5uA - possibly insignificant.
I'm afraid not. That's the worst case specified pulse that happens during WiFi communication when reception is bad. It happens many times, I'm not sure how often that pulse would appear, I'm guessing every few dozen milliseconds. Lets say, 30 per second, and the WiFi is up for a few seconds at a time.
I'm going to need a larger capacitor
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I like the idea though, using a quite large capacitor as an averaging or integrating element sounds like it's going to be more accurate (for measuring overall consumption, not instant current.)