I have some experience with the GameRoom Classics bartop (I originally had the SNK Neo-Geo version, before I reused the parts for my TMNT cabinet). I'll help where I can.
It is true that the power supply is built into the monitor chassis board, and only supplies +5 volts. Also, there is no way to adjust the +5 volts the same way you can on a normal Arcade PCB. When I tested my TMNT board on it before, it would sometimes reboot from low power (even with just +5 volts, that game does run. Just with no sound). For your build, I'd be concerned about trying to run a Pi and an audio amplifier off that (may be too much power required).
To get around that, you mentioned you are open to putting in a second power supply. Power the Pi off a USB cell phone charger. If you go with something like the USB speakers I posted in the other topic, use a USB charger with 2 USB ports (to power both the Pi and the speakers).
The "menu" button is wired to the "Tilt" button on the JAMMA harness. So on your Pi, set "t" to return to the game select menu (if that's how it works on a Pi. I haven't tried using a Pi for arcade stuff myself.)
This bartop does not have buttons or wiring for coins, test, or service. You probably want to add those in (probably put test and service inside the back somewhere.) (The original boards were programmed to coin up automatically when pressing start. Kinda wierd to me, but I can see why they did it. Don't want these boards used in a real arcade machine making money on location.)
Good luck with the mod. That is a painfully tiny control panel. Depending on your woodworking skills, you might want to remove the top part, cut off the sides of the bartop where the control panel was, and build on a wider control panel box. Although if you're trying to preserve the original artwork, that might not work well.
-Edit: Are you planning on mostly playing this by yourself, or do you want it 2 player? You might want to think about removing the control panel and replacing it with a 1-player "Street Fighter" layout. Just a thought. If you do, the control panel top is secured with 4 normal screws. The panel itself is just a thin flat rectangle of wood (just 1/4 inch thick, I believe). Should be simple to cut a replacement. Drill holes as desired, maybe add some art, add controls and wire them up, and reinstall the panel.
Also, be very careful with that CRT tube. It's the same exact type I broke last year, and I could not find a replacement tube to save my life. Ended up replacing the whole monitor (tube and chassis board).