I was thoroughly dismayed at how quickly the 135's got dirty and the difficulty in getting them clean, back when I first bought mine in 2012. When I needed rear brakes shortly thereafter, I knew what to do - go with Akebono ceramic pads, which are virtually clean as a whistle. Yes, their stopping power is slightly less than the dirty stock pads, so a slight amount of additional pedal pressure is required - but other than that, they stop fine, and wheel cleaning issues immediately become a thing of the past. I now have them at both ends so I've been driving clean wheels for years. I can go 2-3 months between cleaning them, and they stay spotless.
Might not be a practical solution in the near term, but definitely worth a switch next time you do brakes.
'21 M2 Competition
'08 Z4 M Coupé
'19 Golf R