Thank you, that's a very helpful response.
After reading your post, I have been reading up on powering the pi from the pc's psu. If I understand correctly, it looks like the 5v supply is for standby mode, which I think might leave the pi in an "always on" state. I think that this in turn would mean that the monitor would stay on as it always has signal, which means when I press the on/off button on the cab, the lights will go out but whatever is on the screen will keep running (?)
The alternative, as you say, is for a multi-socket extension lead inside the cab with plugs for the monitor and the pi, and - I guess - separate transformers for the cold cathode and the 5v (led) coin switch. But how could I then safely connect the on/off micro switch to operate all plugs on the extension lead ?
I really want to keep the arcade button as a momentary on/off switch for everything in the cab. So I think maybe I'll try connecting the arcade button using ETA prime's method on YouTube. But power it from the pc PSU. And to the same button, I'll run 2 other wires to the momentary switch for the PC PSU. I'm hoping this will solve the power issue.