While Arcade is very vocal about his hatred of Ceasar's Legion and any other fascistic type of leadership for the Mojave, it is possible to both recruit Arcade as a follower and complete this quest while serving the interest of the Legion or Mr. House. To do this, you can wear an NCR uniform when recruiting Arcade and then, as the quest progresses, do some clever hiding of the truth until the final conversation with Judah and the Remnants in the bunker. Sadly though, Arcade will flat-out refuse to fight for the Legion, leaving the player at this point.