OK... got to thinking about this, can't really stop.
The way I see it that there's a couple of hard time constraints.
First, as this has to take place after 1783, it has to take place either late in the reign of the Qianlong emperor or immediately after. Qianlong was basically the Justinian of the Qing dynasty; he left the empire at its territorial apex but killed the treasury and weakened his dynasty. Afterwards the Qing went into decline.
Arresting that decline has to take place before ca. 1817 when the British decide to smuggle in opium, which causes a terminal decline in wealth from silver exports and legitimacy culminating in the First Opium war in 1839.
Grabbing a piece of North America and populating it is also a crazy prospect with few opportunities. I think the quickest way for this to happen is for there to be a territory grab followed by a quick mineral rush. The historical mineral rushes were the Alaska gold rush, the Fraser Canyon gold rush and the California gold rush, all of which give us three candidate territories: it can either be Russian Alaska, British Columbia or California. All of these are held by European powers during the time constraints except for California, and that basically restricts a land grab to the time of the Napoleonic wars or to the Mexican war of Independence (1810-1821).
I would rate these to be in ascending order of likelihood. China would not want to antagonize Russia which also happened to be a land neighbor. Even without the TSR that would invite more problems than it would be likely to want to handle. British Columbia would be more likely, but especially to an open China, the British would be a devastatingly bad enemy. No, it has to be California, and the ripest opportunity for taking it is to take it away from the Mexican rebellion.
That leaves us with two problems: first, how to get China the capability and international credibility to do so, and second, how to get the California territories... let's call them 金山 (jinshan, more or less the OTL name for San Francisco) into the United States.
The second bit is easier than the first. There can be one of two scenarios, the first of which is a Texas or a Hawaii situation: Americans go to (a possibly independent, breakaway) Jinshan, become a strong and well-equipped minority, and then expel or displace the Chinese, possibly with the help of US troops and possibly by co-opting the revolution. Alternatively, it's a Spanish-American war situation where the US defeats China in a war and gets colonies. Either way, this would have to either be very early when China is still modernizing and can't effectively project force, or later, perhaps when China goes back into decline and American manifest destiny hysteria is at its peak.
Getting to this particular POD is difficult. You basically need a modernizing emperor willing to drop the bullshit and effect a Meiji-style revolution, at least temporarily, to give China enough power to get colonies. However, the reactionary coup necessary to restore a conservative emperor would not be. Here's what I envision:
In 1792 or 1793, the Qianlong emperor is assassinated by a court faction supporting the accession of a different heir than the chosen prince Yongli during the course of the Macartney embassy. Yongli is able to escape to the British squadron at anchor outside Tianjin, which defeats the war junks sent to capture him, and before the new emperor can call up the banners a mixed force of loyalists and British marines are able to storm the palace and capture the conspirators. In gratitude, Yongli decides to liberalize trade relations and take on British advisors and observers, and has a decidedly more pro-Western outlook. He is now notably also aware of how outdated his army is. Now the Jiaqing emperor, he purges the corrupt administration of his father and recovers significant resources. Combined with port tariffs, he now aggressively studies Western methods and surrounds himself with cliques of modernizing officials while Confucian scholars are sidelined. In 1795 the Miao rebellions erupt; Jiaqing calls up the traditional banner armies which perform poorly. In response he redesigns model regiments of the Green Standard Army on British lines, which quickly and more efficiently break the rebellion and the later White Lotus rebellion. Jiaqing also begins to develop a powerful navy that begins enforcing Chinese interests in the region. Fast forward to the outbreak of the French revolution and their segue into the Napoleonic Wars, China weakly aligns with Britain, but focuses on modernization while sending token squadrons to aid in the capture of Ceylon and Java.
In 1811, China gets a taste for colonies and seizes California from rebellion-wracked Mexico, and preemptively sends a modern fleet to Manila Bay to convince the Spanish, who have other problems, to back down. ca. 1815, gold is discovered and ridiculous influx of Chinese, some of them forced laborers or prisoners, are shipped to the new colony of Jinshan to mine. A minority of Americans also join, and together they largely displace the preexisting population.
Shortly after, conservative elements in the court sidelined by modernization ally with Northern Qing banner army commanders disgruntled by their sidelining under the Jiaqing regime; they depose the Jiaqing emperor and curtail trade relations with the West, with special ire reserved for the Jiaqing's favorites the British, who are unable to intervene but retaliate with opium smuggling and open hostility. The new conservative elements in the court and presumably the new puppet emperor detest their predecessors' openness. China's modernization is halted, the new cliques executed, and the stage is set for eventual decline... but in Jinshan, the remains of the modernized Green Banner-type colonial army, there to present a credible deterrent to the Mexicans and Americans, take power and impound imperial sailing ships, creating an independent Western-facing state with perhaps tens to hundreds of thousands of Chinese forming an overwhelming majority in OTL Northern California, perhaps who turn to the British for help as a protectorate... and the stage is set.