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Summary
1. Install SCP Toolkit
2. Install Antimicro & Configure It
3. NEW STEP: Install & Configure Palette Manager [The Source Is in Section 4 in Here]
4. Enjoy! [See Some Tips in Section 5 in Here]
Addendum
I got into detail on how I make this work for me in this post.
If you're experiencing the issue where your controller is properly recognized in Windows but doesn't appear to control PSOBB, but X and A buttons on the controller or other labeled buttons still work, the issue is likely that you have two or more controllers plugged in or have a driver like vJoy...