I've took "measures" and now I have the Game Kit's shield assembly in 3D:
It's the first step to make a case for the thing so that it looks more "finished". While I've been creating some very few 3D print designs in the past, this one is the first that I'll try to do for something functional.
I'm really tired of the current state of my device and I want a proper casing as soon as possible! Hopefully pressing buttons will become less painful.
Apart from that - you might notice that the USB shield is between the joystick and the screen. This isn't how it was intended I believe, however, this way I can still press the screen buttons - at least two of them quite conveniently. I hope to make use of that.