Only hearing the noise when the radio is receiving is a clue. Try opening the squelch on the radio & see if you hear it continuously. If so, that's a hint that the noise is getting in through the radio, but you still don't know if it's RF (being picked up by the antenna) or riding a ground into the radio's power supply.
Unfortunately, there's no universal cure; just some things to try.
Easiest 1st would be ensuring that the radio's antenna coax shield has high quality continuity from the radio all the way to the antenna, and the antenna ground has a good bond to the airframe.
Next would be trying the recommended filter, but at the LED module terminal(s) instead of the switch. In other words, a filter on each module.
Next would be the LED modules' ground paths. If the filters don't solve it, try a temporary ground wire (obviously, while the plane is on the ground) from each module back to the 'forest of tabs' near the battery. In other words, lift all the LEDs' local airframe grounds & run the grounds direct to the battery negative.
All the above are ways to chase ground path noise. If the LED modules are indeed radiating RF energy into the radio's antenna, that's going to be much harder to address.