It will be difficult to preserve a 'Four Kingdoms' period if one of the conditions is a 'Southern' Ming. By 1644 a lot of the strategic dust had settled: the Qing regency was firmly under Dorgon, Li Zicheng had surmounted the Ming (thus ending any realistic Ming attempts to compromise/placate him), and the Southern Ming would be under the control of relative incapables unable to agree amongst themselves or put up a strong defence against foreign invasion.
I can think of two potential PODs, both requiring internal dissent on behalf of the Qing:
1) 1642-1643. Li Zicheng advances across Henan and is on the cusp of breaking into the Ming Heartland; at the same time, Qing armies become increasingly bold in their incursions around the Beijing area. Ming court compromises with Li, giving him kingship of the Chinese northwest as a means of placating him/getting some breathing room. Li Zicheng, fearing that Beijing is not such an easy nut to crack, agrees. In 1643, Hung Taiji of Qing dies, and Jirgalang decides to contest control of the regency with Dorgon, paralyzing Qing governance. Ming can spare no resources for the various rebellions in the Southwest and these lands eventually fall under the rule of Zhang Xianzhong.
2) 1644. Li Zicheng, having captured Beijing, is quickly defeated by the Qing but manages to retreat in good order to the Chinese southwest. Qing (under Dorgon) decides to focus on Ming first due to their legitimacy/state capacity. Southern Ming gets their act together, but Qing troops overrun Nanjing, Fuzhou and Guangzhou between 1644-1647 while making some inroads against both Li Zicheng and Zhang Xianzhong (who would ideally have the sense not to be massacring too much of Sichuan). However, in 1647 Dorgon of Qing attempts to purge Jirgalang from his position of co-regent in order to replace him with Dorgon's brother, Dodo. Jirgalang, who is leading troops in Guangdong, mutinies against this and leads troops back north to fight with Dorgon. In his absence, Ming loyalists/Qing defectors take back Guangdong and Hunan provinces, while Li and Zhang do the same for their regimes. The Qing civil war is long and ruinous and the Qing emerge from it unable (for the time being) to unite the country, while the Ming use the time to consolidate rather than engage in quixotic attacks against rebels/Qing.
You could also say that a 'Four Kingdoms' strategic situation did exist in the region from 1620-1640: the Ming, the Qing, the Mongols under Ligdan Khan, and Choson Korea.