Again, this is pretty thorough, and I feel that this discussion has been dragging on for much longer than it should have been.
If a unified China successfully embraces Catholicism,
and the Chinese Rites controversy is entirely avoided, I could theoretically see a different Korea more open to Christianity that it had been IOTL. I've stated this in a roundabout way in previous posts, although I had also mentioned that it would be extremely unlikely for the pieces to fit together.
However, I will say that if the Qing's position becomes relatively unstable within North China, which would be necessary for it to have much looser (indirect) control over Joseon, Korea could theoretically invade Manchuria and collaborate with the Southern Ming in order to conquer the Qing. While this had been
planned IOTL, even before the Qing eventually broke through the Great Wall in 1644, it never came to fruition because of the Qing's relative stability within China, while Korean officials covertly informed the Qing court for fear of retaliation, and Hyojong died in the middle of planning. If the Qing had been destroyed due to a major two-front war, I could see the "restored" Ming promoting Christian theology, then transmitting them to Korea. However, Joseon would have been forced to return any temporary gains in Manchuria back to the Ming, while its treasury would have been bankrupt at least twice over due to four-five major wars
and a coup within 60-70 years, which would have been unprecedented in Korean history. In this scenario, Korea would have been thoroughly exhausted to the point where the Ming could have simply waltzed into the peninsula without major resistance, making the OP moot.
Even if Korea still managed to somehow remain independent under a restored Ming, the extreme economic and social cost associated with decades of continuous conflicts could very well lead to extreme political instability, if not an outright civil war within the peninsula, essentially turning Korea into a "failed state" for at least several "long" decades.
In any case, the Korean population had essentially recovered IOTL by around 1700-50 or so, while the population would have at least doubled by 1800. Both were possible because numerous far-flung villages had gradually managed to build up their populations over 100-150 years. If Chinese refugees had managed to arrive within Korea, they would have been limited to Hanseong (Seoul) and a handful of coastal cities, as any "missionaries" would have to spend weeks, if not months, traveling from remote village to village, deterring them from traveling much further inland. Given that the peninsular population would have eventually recovered within 50-100 years with or without Chinese migrations, which would have collectively numbered far less than 100,000, the vast majority would have settled within the capital anyway, as workers within villages were not needed (rather the opposite, as refugees had fled
en masse from urban regions), even after major devastation. A continuing stable influx from abroad would also have severely exacerbated social tensions over the long run, mostly due to potential concerns over "elitism," and would have become a major thorn within Qing policy after Chinese unification.
Southern Ming pirates operating within the East China Sea would also have evoked strong memories of the
wokou for Koreans, and given that the vast majority of maritime routes directly linked South China to Southeast Asia (along with Japan and the Ryukyus), while virtually none were linked to Korea, it would theoretically make much more sense for masses of Chinese emigrants to settle within the Ryukyus (and possibly Japan, if the situation was extremely dire), than for them to gradually make their way to Korea through relatively open sea. Specifically, while a variety of Southern Song individuals (and the navy) did make their way to Goryeo centuries earlier, the vast majority were visitors, not settlers, while the vast majority had emigrated elsewhere, and more Goryeo diplomats had made their way to the Song for over two centuries due to geopolitics.
So yes, a "Christian Korea" is
theoretically possible, but it would come at a very,
very great cost for the Korean populace as a whole.
EDIT: During the Japanese, Jurchen, and Manchu invasions, the devastation within the capital alone was so severe that all of the five main palaces suffered significant losses, and the main one (Gyeongbokgung) was so thoroughly sacked that it was entirely abandoned for over two centuries. As a result, another war would have destroyed Joseon entirely.