Thank you!
I was hoping that there was a way to access this thing from the front (e.g. by removing D-Pad...), but apparently there isn't.

I tried canned air and also folded a piece of paper two or three times, drenched it in isopropyl alcohol and used it to clean the gaps next to the rotary controller.
It looks like that helped a little, so for now I'll refrain from disassembling the device.
But I fear that day might come eventually... ;)

Sebastian