OOC: I think whether Koryo can mobilize 200,000 soldiers or not, there would still be a possibility a Mongol-Japanese army could conquer them in times of turmoil. I think the real question is national identity, but this was a non-issue in the 14th century. Think about it, in this world, the Yuan Empire conquered a similarly militarized but significantly larger country (Japan) than Koryo in a naval invasion, is it really impossible that Koryo could fall? There is always an element of chance, and here the "Nipponese Khanate" is just incredibly lucky.
OOC: The Yuan managed to invade Japan IOTL with a navy that had been built mostly by the Chinese (Jin & Song) and the Koreans (Goryeo). Most of the ships had also been hastily constructed, leading to low quality, and was part of the reason why they were swept away by the typhoons, although the shape of their hulls was another factor. However, once the Yuan is forced to flee north, it will essentially become landlocked, and even if it does manage to somehow retain areas like Shandong and Liaodong, most of the shipbuilders will be located in Southern China, not to mention that the Mongols will be forced to deploy the vast majority of its ships against China, let alone invading Korea.
The government in Japan would also take advantage of the chaos in China to declare independence, as diplomatic relations would essentially have been severed, and would have been much more concerned about internal stability to even think about invading the peninsula. In any case, Goryeo will either continue to pay tribute to the Mongols and retain close ties, as Yuan and Goryeo rulers had remained as blood cousins for a century (up to Gongmin, although he eventually decided to revolt in order to firmly assert national sovereignty), essentially eliminating the need for an invasion, or ally with the resurgent Chinese state in order to drive the Mongols back, similar to what had occurred IOTL.
It's also important to note that the Mongols never succeeded in conquering Ganghwa during seven attempts over 26 years, despite recruiting former Jin shipbuilders, along with numerical superiority, while Goryeo actually managed to recapture its former northern territories and temporarily expand into Liaodong after the Yuan's fall, despite numerous raids by the wokou throughout the period. In addition, during the Imjin War, Joseon (with Ming support) managed to repulse the Japanese after tying them down within various regions, and despite the fact that virtually the entire navy was constructed from scratch since 1592, the Koreans suffered only one naval defeat during seven years of warfare.
As a result, while an invasion (if it occurs) might force Korea to pay tribute in an extremely unlikely scenario, outright conquest and occupation would be virtually impossible.