Long story short: Would the Chinese know about the thing we call the Dark side of the moon? Probably not.
If we assume that Disney's Mulan takes place around the same time as the legendary character "Hua Mulan", that would be between 420-589 A.D. At that point in time, most of the observations regarding anything close to "the Dark side of the Moon" were by Anaxagoras (510 - 428 B.C.), Plutarch (46 - 120 A.D.), and Jing Fang (78 - 37 B.C).
Anaxagoras, a Greek Philosopher, proposed that the moon did not emit light but simply reflect light from the sun. Plutarch, another Greek Philosopher, wrote that the moon had spots where the light of the sun did not reach and that certain spots seen on the moon were just shadows of rivers and lakes. While neither of these Greek Philosopher's hypothesize directly about "the Dark side of the moon", both observations, especially Plutarch's, are alluding to spots like a "Dark side of the moon". Is it totally possible that some of these writings made it over to China by the time Mulan was around, but even if they did, neither one's writings directly mention a "Dark side of the moon".
Jing Fang (78 BC - 37 BC), a Chinese mathematician and astrologer, observed the basic phases of the moon, proposing that the moon and sun were both spherical in shape, and there is a quote where he alludes to dark parts of the moon. The quote is as follows: "The moon and the planets are Yin; they have shape but no light. This they receive only when the sun illuminates them. The former masters regarded the sun as round like a crossbow bullet, and they thought the moon had the nature of a mirror. Some of them recognized the moon as a ball too. Those parts of the moon which the sun illuminates look bright, those parts which it does not, remain dark." This would later be a part of the "radiating influence" theory. This theory was dismissed at first, but later embraced by Zhang Heng (78 - 139 AD).
So all of that being said, in China during the time of Mulan there had already been quite a few observations about the moon, so it's not out of the realm of possibility that there would be legends and stories about the mysteriousness of the moon. The most famous legend is that of Chang'e, but the legend doesn't include anything about the Dark side of the moon.
Edit: to clarify the last part, while is not out of the realm of possibility that there would be a folk legend of some sort about the dark spots on the moon, there isn’t any evidence suggesting that the Chinese knew about a place called “the dark side of the moon”. As noted above the only moon legend I could find is Chang’e. It’s difficult to put an absolute “No” on something like this simply because China is so large and so diverse, however, if we’re strictly talking about the far side of the moon and if ancient China collectively knew about it during the time of Mulan, then the answer would have to be more than likely no.