Sad to see this fork of SC Controller is now archived. It provides an Appimage version, and also worked with my PS4 controller. Credit to Kozec, the original creator.
https://github.com/Ryochan7/sc-controller
There was also Qjoypad, but I haven't used it in a while.
Basically, the game is reading the raw controller input as well as the translated virtual controller input. I've run into that a lot before on other games. The fix I found is usually to try another mapper, or to disable the controller in the game and map the controller to keyboard and mouse. It's annoying.
With luck, you -might- have luck with closing the game and setting up the controller mapper, then start the game. If the mapping program provides an Xinput option, try toggling that and see if it helps.
Good luck.
EDIT: Did Kozec stop developing the app? I used to support them on Patreon before I lost my job. There haven't been any official updates in a while. Sad day. :(