Update: I achieved what I wanted to do! 
I modified the WLED 0.13.3 code myself in VSCode with PlatformIO (I was having issues with the 0.14.0-b1 for some reason).
In the wled00/dmx.cpp file, I defined two arrays of numbers equally spaced between 0 and 255, one for the TILT (vertical) value of the light, one for the PAN (horizontal) value. They basically represent the angles I want the moving head to get to. For example, I wrote 8 values for the PAN, so that there’s a position every 45°.
In the same file, I added two new switch cases that each pick a random number in the corresponding array, and send it to the DMX output, just like the existing cases.
Finally, to be able to select these options in the WLED DMX output menu, I added two new “option” tags (“TILT random” and “PAN random”) in the “select” tag of the element with the id “dmxchannels” in the wled00/data/settings_dmx.htm file. The value of each tag is the corresponding integer in the switch case I talked about above.
Once the firmware is compiled (binary & HTML!), flashed on the device, and the DMX channels are configured, everything sould be working!
The effect is very similar to the embedded modes of the moving head light, selecting random positions over and over.
It’s not synced with what the lights are doing, but since I use my WLED device with LedFx, the leds of the moving head are already synchronized with the music, so there’s no need for more. It’s exactly how I envisioned it. And the motor is not fast enough to follow the rythm most of the time anyway.
If you need some help achieving something similar, don’t hesitate to ask me, I would be happy to help, maybe even provide you my modified version of WLED!
One last thing: before doing all that, I did tried to map the RGB values of WLED to the TILT and PAN channels of my light (as @packetbob suggested). It was working more or less (depending on the colors of course), but it wasn’t using all angles, and most of all the movement was quite stuttery (it looked like the moving head didn’t know where to go, which was quite funny). That’s why I tried to do my own thing.
