I think I've found the issue. There are two connections on the button as well as ground. I've switched few of the buttons over to the second connection and they all now function as expected. The ones I haven't changed still result in continuous firing.

I'll continue to rewire and report back if the results are good.

I managed to get it to work!

Turns out that I had to change the retroarch.cfg files inside the individual emulator folder, since the global retroarch.cfg file changes did not seem to register for the Neogeo ROM :)