I haven't used Hyperspin beyond trying it out back in the day, but it only doing that in one wheel and not in game would lead me to believe that it is caused by some other controller (such as the trackball or spinner) mapped in that wheel that is not mapped in the others. I'd unplug the trackball and see if that makes a difference. If not, unplug a connector from one of the terminals on the switches one at a time. If it stops, you've found the bad switch. You can just replace the switch for cheap.
They look like standard Happ/IL sticks. Needing to cut the corner off is going to prevent you from using most options with a balltop (pacman reunion) because they have a square housing. A quick search came up with IL Eurosticks with ball tops, but they were from vendors I am not familiar with.
You can get shafts that allow you to switch the current joysticks to balltop:
https://focusattack.com/spare-parts-tools/popular-brands/industrias-lorenzo/eurojoystick/(Although those are IL shafts, they should work on Happ as well.)
P.S. I hate that button location and the collars that raise all the buttons.
