Some monitors actually have those household plugs. If you want to use it you can cut the plug, split the lines and splice it into your input line. If it has a household plug, that means it doesn't need the iso-trans. You can totally by-pass it.
The plug with the 3 wires that you've got in the cabinet is most likely the power for the old monitor (which would have required the iso-trans and would already lead into it if it's wired properly).
Without seeing the wires, it's hard to see where they go, but you SHOULD be able to cut the monitor power's plug off, split the wires, and splice them into the 2 purple wires leading into that plug in your cabinet. Using an iso-trans when you don't need it is acceptable.
The other fool-proof solution is to chop the monitor power plug, split the wires, and feed them directly into the power cable coming into the machine.
Lastly, if you're concerned about chopping the monitor's power plug... Plug the arcade machine and monitor into one power strip and plug that into the wall.