Here in the U.S., the burden falls on the orthodox Jew to arrange their own life so that they can observe that stricture.

And I would agree they should. The example I was thinking of was a news article about those folks trying to fly. One did exactly what you said and wrapped himself in clear plastic with the face open. They actually had a photo in the article. But another caused a fuss because he would not take his seat (as the flight attendant instructed), and was insisting another person change seats with him so he would not sit next to a female. I don't recall what happened (I think he may have been ejected from the flight for failure to obey instructions) as I read that a year or two ago. But I think he was wrong to expect people would rearrange their seating for him. Regardless of what I think of that stricture, it was on that guy do adapt to the situation. Not everyone else to accommodate him.

I don't know what the law has to say about that, other than what you wrote. And I don't know how airlines are expected to handle it.

Also, as a point of interest, since it revolves around menstruation, could he ask "are you on your period", and if the answer is "no", could he then sit next to the woman? Yeah, that's a question that a woman may not want to answer. But if she did, would he be allowed to sit?