Yesterday I opened my GCW again and put back the original GCW Zero membrane under my D-Pad. And now I think everything is fine. The controls feel precious, the diagonal inputs work fine and I am also able to do Special Moves and Hadokens, Shoryukens etc in Street Fighter. The 3d printed D-Pad is still far from perfect for Street Fighter games but for most other games it's good. Overall I am happy with it now :-)
By the way: I don't recommend the original Gameboy rubber membrane for the GCW Zero D-Pad. The inputs feel too hard with it and I also got problems with diagonal directions. But it's good enough to use it for the GCW Zero Buttons. The buttons feel really good with it.