You'd need to screw China pretty badly to get them to not demand and receive Jiandao (a POD at the Gando Convention would see the Chinese viewing that region as as Chinese as the rest of Manchuria, meaning conflict and it's not like Korea, even united, can stand up to Chinese manpower). Also not sure how getting part of Manchuria butterflies away the Partition, seeing as the Joseon probably end up in Japanese control anyways (nothing too big has changed at that point) and Japan's not going to be able to beat the USA and Russia/USSR (because their interests naturally conflict, Communists likely take power in Russia, still split Korea).
Korea getting more productive crops to fuel a population boom and naval advancements seem a more obvious way to help Korea, unless Japan, China, and Mongolia are gimped hard. Land expansion's only going to lead to further land conflicts, which no Korean nation since Goguryeo has really excelled at, while the Korean naval tradition has been around since the time of Three Kingdoms. Manchuria's land isn't rich enough nor safe enough to ensure population growth (if anything, they'll just get invaded from the steppe more) while the peninsula has been comparatively easier to defend. Think Venice as a merchant republic vs Venice as the would-be conquerors trying to unify northern Italy only to get stomped during the War of the League of Cambrai. Taking a Portuguese/English/Dutch approach would suit Korea's geographical position and economic health much better than launching more northern campaign in a neverending fight with nomads (neverending since the Chinese had been fighting off raid since there was China).
My thought is with the Gando Convention as the POD, the Japanese claim Gando as Korean land, and we go through the 20th century Korean history from there. The USSR and US respect Gando as Korean, and it's added to the DPRK upon partition. If the second half of the 20th century and the 21st century is conducive to it, then Gando along with the rest of North Korea unifies with South Korea into one Korea.
Manchuria, rich enough? Chinese were pouring into Manchuria despite regulations against it during the Qing. With the potato, Manchuria is very conducive to agriculture, not to mention with buckwheat, already a Korean crop. I'd love to mention quinoa too, but that might be for a more exotic TL where quinoa is spread outside the Andes.
Manchuria has rivers and defensive boundaries which would allow a limited Korean expansion into the region. Not to mention, Vladivostok, the premier port in the region, is very near Korea itself, just over 150 kilometers from the border of Korea which has changed little since the Joseon.
But Manchuria was liable to invasions by steppe hordes, which is what led to Manchuria becoming forever lost to Korea in the first place (Liao, then Jin, then Mongols, then Manchu/Qing). The Goguryeo were fighting Chinese dynasties, not exactly the most viable strategy for a long, prosperous history.
Also, Mt. Baekdu is going to erupt in the era (it's theorized that helped lead to Balhae's collapse and conquest by the Liao) since geographical PODs are ASB or something? Can't butterfly away natural disasters like volcanic eruptions and such a natural disaster will make holding Manchuria, well...
As for Vladivostok, Russia likely would've tried seizing it for the sake of a warm water port and getting the trans-Siberia railway to the sea, whether it belonged to Qing or Joseon.
IDK, just Korean track record for the last millennium and a half doesn't really show much of a military powerhouse so much as a nation outpopulated by all of its neighbors, a fact that would not change all that much with just the acquisition of Manchuria (which would be near impossible to hold for long, just too far to the nearest defensible border, too many enemies).
Yeah, Mt. Baekdu will erupt, and destroy the Balhae in the process, but it isn't like Mt. Baekdu has erupted since. Which might place a limit on the earliest date for Korea to seize even a portion of Manchuria, since before then Mt. Baekdu will wreck the state in charge of the area.
Russia can try, but it isn't like they'll automatically be successful. If the Koreans partner with China, they can rebuke Russian influence in the area.