I can appreciate setting an early POD in order to give them the best possible chance, but at the same time, how would you actually achieve this so early? What would be the catalyst?
What I've thought up to was this:
I have finished up the 1860s. You could see the "planned timeline" in page 1 or 2. From there, however, Korea does several major things: I) start of intervention in Japanese politics II) gaining sphere of influence through Taiwan, Hainan, eventually Vietnam and Taiping Republic. As the first Asian nation to have, perhaps, successfully modernized, it is seen by many Asian countries as a model to follow. Using this prestige, Korea quickly gain influence in Vietnam, Hainan and Taiwan. (Taiwan and Hainan becomes independent in the Second Opium War.) From this time, Japan is torn between Korea-associated, US-associated and Russia-associated factions within the Shogunate court.
Anyway, when Lee Ha-eung abdicates his throne to the son, Gojong, the first thing he does is to proclaim Korea independent from the Qing dynasty. Although it has lost half of its territory from the Taipings, the Qings have had some modernization in the military through the late 1860s into the 1870s; therefore, confident in their success, China proclaims war against Korea. However, after 6 months of fighting, Korea emerges victorious; during the process of battle, Korea forces Taiwan to become its protectorate and uses it as a military base against mainland China. This is only possible, however, because of several major instances- France was busy recuperating from the Franco-Prussian War; Britain was preoccupied with the Great Game against Russia in Central Asia. Therefore, all are disinterested when Korea annexes Taiwan in 1877. Soon, Taiwan becomes the core element of Korean colonial expansion.
Koreans also plays a major part in the Franco-Vietnamese War of 1882 as military supervisors and helping Vietnam win against the French.
With deep relations between Korea and Prussia dating back to the 1830s when Koreans first arrived at Prussia, Korea-German cooperation is well seen during the Scramble for Africa; due to much German help, Korea gains Tanzania as its ultimate and only African colony. Germany also gets Korean help to establish a colony in China- Tsingtau.
On the other hand, Japan, much weakened by fights between samurais for power as Shogunate, serves as a military base for the Russians, whaling port for the Americans, another Hong Kong for the British and sphere of influence for the Koreans. The OTL Choshu Five, core elites who were to lead the Meiji Ishin, remain frustrated in their efforts to unify Japan as a single nation.
During the 1890s into the 1900s, Korea becomes much more prosperous due to the increase of export of products to China and Vietnam. Due to decreased French influence in Southeast Asia, Korea becomes much more aggressive in North Indochina, such as planting Koreans in key jobs within the Vietnamese military and government; another forced policy was to accept a modified version of Hangeul as the new writing system for Vietnamese. This was reflected from the French policy in Cochinchina-Cambodia of romanizing the Vietnamese language.
The French are angered by their reduction of influence in the Far East. So was Russia with its influence in China kept in check by both Britain and Korea. In 1900, when the Boxers were subdued or killed, Russia furthered its troops south and did not remove them; on the other hand, France began to station large numbers of troops around legations in both China and Taiping Republic. A start of war was shown when a coup in the Republic of Hainan led by a French-supported faction became successful.
Claiming "atrocities upon humanity" caused by this faction's coup, such as the successful murder of the Korean ambassador and his whole family, Korea found it to be causus belli to start the Second Franco-Korean War in 1904. While Russia also jumped to the French side, both were underequipped, undermanned and unprepared to start a war with the greatest nation of the Far East. Although the results may have been less drastic if France and Russia were in better circumstances to start war with Korea, the result within the Second Franco-Korean War (or just the War of 1905) was that both nations lost badly and were forced to agree on Korean victory.
With the occupation of Hainan in 1904, Korea had a tacit agreement that, while the United States occupied the Philippines, Korea was to occupy Indochina; therefore, on 1910, there was no opposition except the Vietnamese when Korea formally annexed Indochina.
Now, that is up to 1910- I should start fresh from there.
That was what I thought to be a basic format. Of course what I wrote above is heavily flawed, so I dont mind criticisms or even nitpicks. But, of course, compliments would be best.
Does that even answer your questions, I don't know but I hope so.
I think the "catalysts" you mention may be implied somewhere in this TL. For example, Korean students have gone to France since the early 1810s; however, in the chaos of 1830 in France, many Koreans go to other places, such as Great Britain, the Netherlands, and Prussia. This Korea-Prussia relation that starts from here becomes useful much later, as shown. The "timeline" above was just straight out of my head. Please criticize, admonish, chastise, whatever you wish.