Hi, I just tested to put three 5,5V 1.5F ultracapacitors parallel and connected the system to the APM power rail. APM stayed alive 10 seconds when normal LIPO/ESC was disconnected. Telemetry radio was also connected to the APM and transmitting full power.
During normal LIPO battery usage the VCC looked very steady even if I made some heavy servo and ESC throttle movements every 10 seconds for 5 seconds each time. The capacitor system weights 14 grams and it is not much bigger than a five cent coin. It powered standard 2.4GHz RX about 5 minutes (no servos connected).
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Yup, I've played with this too. I have used a large cap bank from HK, and have also soldered SuperCaps directly to the board.
A word of caution though. It can cause problems on the bench. While testing it, I've had it where I had the battery system connected right through to the motors. Then I would disconnect the APM power supply to test it. The motors suddenly started. The problem is that when the APM voltage goes below 4.5V, it gets "undefined" and weird things happen. The SuperCap holds it in this undefined state for several seconds.
So be careful what you're doing with bench testing.
But I'm surprised at the level of noise you're getting there. It seems kinda high? I wonder if the ESR on the supercaps is too high?