19th century is too late, barring massive PODs. Whereas Japan had been in contact with the West for centuries through Dejima, the Joseon were a true hermit kingdom that refused any contact with Europe and America until the 19th century, after the Qing were discredited by the numerous unequal treaties. Not only that, their population was quite a bit smaller than Japan's and the yangban controlled everything (one of the kings was an illiterate puppet). They're just too behind to overtake Japan without Japan being terribly screwed, are at risk of absorption by the Russians, and even if the prior two conditions go their way, they'd be outshone by China, fragmented or otherwise, very quickly.
The best POD, I'd think, is to kill off either Seonjo or Injo years before their OTL deaths because Gwanghaegun had the best chance to keep the Manchu from invading twice and keep good relations with the Qing and Japanese and Injo's' eldest wanted to bring over Western learning and medicine (the former was overthrown in the coup that crowned Injo and the latter was either poisoned or bludgeoned to death, probably by Injo, his father). Those two managed to get the Joseon invaded 3 times in the span of 30 years in wars that killed something like 10-20% of the population through famine and started the Joseon isolationism that only ended in the 19th century.