I hope it never becomes an issue. It's unlikely that this particular mountain is the only place on the Moon where it's a good place to set up shop, and it would be difficult, and supremely stupid and futile, for ANY country to try and keep some other country from occupying an acre or so of the Moon's 14.5 million square miles. It's kind of like Balboa claiming the Pacific Ocean.

Frankly, if China gets there first, and sets up mining or anything else, they are welcome to use as much of the Moon as they can afford to. The treaties say nothing about individual entities being able to claim and use a chunk of the surface. I wish them luck. Getting to a polar landing site is no piece of cake in terms of delta-v.

Now, if they try to shoot down others in space trying to land somewhere else on the Moon, that's a different story. I hope it doesn't devolve into territorial battles, but as far as history goes, it sure wouldn't be the first time.