The badge may be housed in a 3D printed case, described below. The case consists of three parts: a bottom lid, a main badge case and a tiny manhole cover. The main badge case benefits from some supports.
The STL files can be downloaded directly here or from Thingiverse.
You can use support blockers in your slicer software to only have supports in the center of the main badge case. This should provide enough support for your printer to bridge the wall.
The badge has slots and cutouts for the button, traffic light and QT Py ESP32-S2's STEMMA cable. If you don't want to use the STEMMA cable, you can cover the hole with the tiny manhole cover.
You can print the parts in white PLA so that you can paint the different details of the badge. Acrylic paint works great for covering PLA.