I think bindis are a bit of a toss up. They are NOT Middle Eastern, but you will see them worn by non-professional dancers enough that you're not likely to be yelled at for it. You could make a very small case for the overlap of Middle Eastern culture and that of India due to the spread of Islam, but I don't know how convincing it would be
I've also seen women wear bindis as eye accents which somehow seems to work better since it doesn't put two cultures in as direct a conflict. The picture I attached is from my first American Cabaret performance. I still enjoy this costume (though I've had to alter it since I'm not quite
that tiny anymore!)