It used to take my a long time to clean dispersion screens. Scrubbing, toothpicks, etc. I finally took them to work and cleaned them in an ultrasonic bath. Eventually I bought one for home and found I don't need an industrial strength version. Here is the one I purchased. https://www.amazon.com/gp/product/B0006DOALG/ It was only $80. This exact one is not currently available, but there are lots of similar choices.
Here is the "before"
The basket holds everything. Even the portafilter, keeping the handle out of the water:
Add a dash of Cafiza and 6 minute runs takes care of it.
Final result: (Brass heat sink is still a little tarnished, but no coffee residue)