You can replace the large bulk capacitor with a regulator with a fast control loop - meaning you don't need to store as much energy, the regulator takes care of keeping the output correct, and can do it by having extra input voltage available (which is allowed to dip more). In practice, this may mean modern fast LDOs, that can easily respond in megahertz range. Most switchers can't, because the switcher's response is inherently limited by its inductor.
Although, a modern high-frequency switcher should be able to do it as well. I mean, the cheapest, most integrated, modern one-chip buck running at, say, 1.5-2MHz and with a tiny inductor - and hence, relatively small capacitors as well. You shouldn't need 220uF...
This single positive peak with fairly slow rising edge (100us) is bugging me. It doesn't look like inductive ringing to me, but like the device suddenly stops consuming current for a while.
Can you post a schematic, or even better, an actual photo or layout picture that shows the full regulator, it's output caps, any filters you have currently in place, and finally, the bypass capacitors for the load?