Your proper remedy is limited to vacating the arbitration award. Which is precisely what in fact transpired here.
That said -- and that's all that need be said to resolve the case -- I think that if you're a proposed arbitrator and your girlfriend is the sister of defendant's lawyers, you need to say more than just: “[Defendant's lawyer] is a personal friend. While I do not believe that our relationship would prejudice my hearing of this matter, this relationship does present a potential conflict."
How about: "I am in an ongoing romantic relationship with the sister of defendant's lawyer, and also know counsel personally. Let me know if you would like to select a different arbitrator." Or, even better: "I unfortunately cannot accept the appointment, as my relationships might create at least the appearance of impropriety."