I don’t think they can “win” the Manchuria region short of the Jin Khan dying and thus starting another succession Civil War.I do hope Japan and the Amur Khanate wins the Manchuria region as a way to make Japan to focus on Russia that would be costly in the long term and Joseon would focus on further reforms so they could Japan in the future.
ultimately it might be better if the Amur Khanate is defeated and absorbed into the Jin Khanate. with Japan coming to a new agreement with the Jin against more pressing boarding threats like Russia, Korea, Mongolia and China.