I'm starting to think that it may not be as bad as we think... afterall this is effectively a Gamecube that plugs into this arcade machine. You need to learn more about the capability of these cabs. This is not the forum to do it though. You'll need to join several good Arcade cab websites.
Find out exactly what is inside your cab in terms of any adapters / wiring harnesses / power supplies etc. Take some high resolution photographs to put in your threads and get asking... you never know someone on an Arcade forum probably has done this before and has a thread to point you.
This company sells all the parts you're likely to need including 6-button parts
You might be able to part-ex some of your stuff plus the owner is likely to know everything there is to know about your cabinet and how to make it work with a PC.