Better earlier than later then. Have an alliance of revolting peasants and anti-Manchu townfolk (likely along Yangtse and Pearl rivers and the coast?), i.e. a non-crazy Taiping, break free in the South, while the Qing keep the North. (The Classic Chinese partition.) rivalry will inspire military modernization. Surely Western powers will participate in the game, but if the two powers are strong enough, they may well play these foreigners against each other following their own interests, too.
If you want to remove the Qing entirely, have a moderately successful Southern campaign (but not enogh to subdue the North completely) incite a Rebellion along with a palace Coup, creating a new dynasty in the North, too.
Having everyone "enfranchised" is the tougher challenge.