Let's start with the basics.
How is it wired?
What is the current behavior?
What is the desired behavior?
I assume you are using some software tool to turn the inputs into actions? Or have you just configured the gamepad to handle shifting in game?
Typically with a manual gear box you don't go into neutral by taking no action, you go into neutral by taking it out of gear, and could theoretically get it into any other gear. Paddle shifters are a little different, and if you are trying to emulate that, then I'm not sure why the gamepad circuitry wouldn't suffice by just clicking quickly through the gears.