The reverse rim is the main giveaway to me; the "wire rim" is a classic symptom of being forced through a ring die.
Secondly, of course, is the non-existence of this particular coin in the catalogues. Specifically, this coin is a "mule"; the obverse is dated Year 21, but the "junk dollars" issued in Year 21 are all supposed to be the rays-and-birds variety; all coins with this plain-sky reverse design should be dated Year 22 or 23. The fake-makers frequently make "mules"; partly it is laziness in not matching up the dies properly, partly it is in the hope that someone who knows this coin "should not exist" will think it is therefore a rare pattern or mint error.
If you want some evidence, here's an easy one: Magnetics. A simple fridge magnet will tell you if it's steel or not; many of the cheaper copies are steel. If you have one of those super-powerful rare earth magnets, you can compare the reaction it gives to your coin with the reaction given by a dollar-sized coin you know to be solid silver.
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Despite the fact that it was in the museum, I still have my reservations about its authenticity
It shouldn't surprise you at all; rather, I would say, "because it was sold in a museum, I assume it's a replica". Museums rarely sell genuine antiques or antiquities, and very commonly sell replicas and reproductions. Museums in the West will likely tell you they are reproductions when they sell them to you, museums elsewhere in the world are less likely.
No Chinese law was broken in selling it to you; it is perfectly legal in mainland China to make and sell counterfeit pre-Communist coins.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis