Without knowing what the inside looks like, I can’t give accurate advice. Can you take pictures of the inside of your box? I don’t really know what the inside of that arcade stick looks like.
The easiest way to turn a 4 way stick into an 8 way is to just rotate the square gate. By easy, I mean “smallest investment in parts.” But this will most certainly require some drilling at the very least and possibly plastic/solvent welding stuff.
If you can’t mod the gate, there’s always the actuator. I turned my SuzoHapp joystick from a round gate to a square gate that way. But that was tricky and time consuming…
You also mentioned that you HATE square gates, which makes your remaining options pretty bleak if you don’t want to spend any money. Cutting the gate to shape could be possible, but you’d have to be very, very careful, and that might not be an option depending on how you’re set up under there.
If the gate just bolts on, then you might even be able to cut one out of wood. I’ve personally never tried it, but don’t see why it wouldn’t work.