Ok, my order came through pretty quickly but I didn't have the time to try it out until today.
So far I've tried 100pF, 470pF, 1nF, 4n7F and 10nF. None of them really gave me a flawless transmission but I'd say it was a little bit better than before. Not much change from changing caps values.
I've also tried placing three of the ferrites in series with each anode of the LED before the resistor. Also just one ferrite in series with the cathode of the LED to GND. Both of those gave me roughly the same result as the caps.
I've also tried a mix of caps and ferrites.
What else can I try to reduce the interference? Should I continue up in capacitor size until I hit it or are there other approaches?