User Trehek posted the ultimate Solution to fixing problems with the XInput Controllers, nameley Xbone and Xbox 360 Controllers
There exists a utility called XInput Plus. It is a japanese tool that can be used to change the Output of the Controller. Imagine it like XPadder or such tools, but changing your Gamepad Inputs. It works via modifing and dropping a Xinput.dll in the starting directory (with the exe) of the game. It's similiar to stuff like ENBs. A more detailed tutorial and the way to download it is here:

Warning: Your System may recognize this file als a trojan. Please see the troubleshooting section for more info!

https://community.pcgamingwiki.com/files/file/2034-xinputplus-ver-415/

Be sure to download the Version V4.14.3.

What you'll want to do is point the utility to the OPPW3 executable and go to the utility's DirectInput tab. Enable DirectInput with the checkbox. Then click on Customize and make the following changes to control assignments:

Left Stick-X = X_Axis
Left Stick-Y = Y_Axis
Right Stick-X = Z_Axis
Right Stick-Y = Z_Rot
LT = Button 7
RT = Button 8
DPAD Up = POV_Up
DPAD Right = POV_Right
DPAD Down = POV_Down
DPAD Left = POV_Left
A = Button 3
B = Button 4
X = Button 1
Y = Button 2
LB = Button 5
RB = Button 6
BACK = Button 11
START = Button 12
LS Button = Button 9
RS Button = Button 10

Click on OK and then Apply and the utility will generate the settings files. Leave the utility open and try launching the game. If you hear a bell sound upon startup, the settings should be working. The Start button should skip cutscenes, A and B should be OK and Cancel in the menus, sticks should work, all buttons should work and be rebindable in-game.

Once you confirm that everything works, exit back to the utility and go to Settings->Others. Disable the "Enable Init Beep" option and click on Apply. Now you're done. You can close the utility and you won't need to run it anymore when you play.

Of course, if you need another button configuration, due to your controller responding differently, you can change that accordingly. That is just the one that worked for me and Trehek.

Tool doesn't work

If the tool doesn't work for you there is a second way of using the tool. Procede like described, but instead of opening the game the normal way you use the XInputPlusLoader in the loader directory. Start it up and set the mode to Create ShortCut. After that navigate to your One Piece 3 Pirate Warriors directory and simply drag and drop the game exe "oppw3.exe" onto the open loader window. It should tell you that it created a shortcut.
From now on you should run the game via this shortcut. Try it and your start button should work to skip the movies and also the triggers should work as intended.


My System says it's a trojan

At least Windows Defender sees the XInputPlus.exe as a Trojan called Maltule.C!cl. This is because this tool uses dll injection to accomplish its goals. Here is a VirusTotal Report:
https://www.virustotal.com/en/file/80b5318af37f1b0b1d0a1c7b8b5a84a09d51c8da66e301560e00783863dc3786/analysis/

Yes it shows one red mark, but all the mayor customers don't see it as malware. Of course, if you don't want to take this risk, it's up to you.