I’m worried about allowed Sharia-based arbitration. Imagine a woman in a Muslim community under such arbitration, and she’s being screwed due to misogynistic Sharia implementation, and wants to go to the courts to get a fair deal as is the right of any American citizen. Sorry, she’s stuck with Sharia.
These arbitration cases don't operate in a legal vacuum. Arbitration, even with a Sharia-based arbiter, cannot be in violation of US public policy. It gets complicated, and I really don't have time to get into it, but there have already been cases (based primarily on Jewish contracts) that arbitration awards have been undone because they've violated protections against sexual discrimination in US law. Jewish law, like Sharia law, also has elements that can be fairly unfriendly to women, as well.