Is September a good month to climb Mt Huashan?

Yes, it is during the best timing to climb the mountain. There would be many sunny days during that period, when you can see the great view of sunrise. And putting up a tend on the east peak is allowed. It is a good idea! Moreover, I'd tell you that although in Sep. it can be very cold on the mountaintop, so prepare warm clothing or rent one on the spot. Good trip!