What is the best recording setup for this game on Xbox 360?

Most people use a capture card to capture gameplay from their Xbox 360 and put it through the OBS program. This is the cleanest way to record footage and nowadays you can buy very cheap capture cards in exchange for minor quality loss. You also may need to buy a HDMI splitter to see the game on your TV/monitor alongside displaying it in your capture card.

Alternatively you may wish to use Xenia emulator to play the game and record that on OBS instead without buying a capture card

Thank you so much Slysonic!