The Manchus that later founded the Qing Dynasty invaded only about 20 years after Hideyoshi did. While they suffered initial defeats against a more heavily armed Ming army, political intrigue and domestic instability (by rebellions) destroyed any chance of an organized response by the Ming. It was thus impossible for the Ming to stop the Qing, as they had no organized armies capable of dealing with the rebels as well.

Conceivably, a decisive Japanese victory against a Ming army in Korea could have achieved a similar result. The Japanese army was far more modernized than that of the Manchus (due in no small part to the sheer amount of firearms that they were able to deploy, although the Manchus were able to eventually develop an artillery force after capturing Ming siege troops and equipment, which the Japanese did not develop). The main problem with the Japanese invasion was logistics, which the Manchus had a reliable supply chain, and was just as if not more organized than the Japanese, with the Manchu "Eight Banner" system.

Hideyoshi initially wanted to use Korea as a base for his invasion of China. Had the Joseon King agreed, there would be a good chance for the Japanese to inflict defeats on the Ming army to the point that it could destabilize China. As it was, the main reason why the invasion of Korea failed to begin with was due to Korean naval superiority in the region, which is most commonly attributed to the Joseon Admiral Yi Sun-sin (who probably has gotten way too much credit, but the man was definitely a naval genius). A docile Korea would not have been able to interfere with the Japanese transfer of troops from the homeland to the mainland.

The only thing Hideyoshi would have to be careful about is intrigue at home. While the Manchus were able to unify the tribes much like Hideyoshi did, Hideyoshi ended up making compromises to pacify potential threats to his rule, most notably to Tokugawa Ieyasu. To appease Tokugawa, Hideyoshi gave him a significantly well-developed and wealthy territory, and Tokugawa had a relatively minor participation in the invasions. When Hideyoshi died, the lords that had supported him were exhausted by the failed invasions, allowing Tokugawa to take control from the Toyotomi clan.

The invasion of China idea was actually Oda Nobunaga's. Had Oda avoided his assassination at Honnouji, he would likely have had more success in his invasions than Hideyoshi, partly due to his superior leadership (and his personal friendship with Tokugawa notwithstanding) but also due to a greater willingness to adapt to new techniques and tactics. For instance, the use of several European-style carracks (ala the Red Ships) could potentially have been decisive in the naval battles around Korea.

Hideyoshi's dream was unfeasible without modifications,he wanted to carve up the Ming empire into fiefs for his daimyos,grant the Japanese emperor Beijing as well as 10 provinces and continue on conquering India as well as the lands to the west.

The biggest issue would be logistics though I think its self explanatory.

Hideyoshi attempted to rule Joseon lands with Japanese administration only to his dismay as it only resulted in militias destabilizing the region.

Hideyoshi wouldn't have made any attempt to coexist with the landed gentry in Ming China therefore he would have no legitimacy and invoke the wrath of Chinese rebels.

Though the Ming army was waning by Ming Shenzong's reign the dynasty was far from keeling over,though the Imjin Wars extracted around 20 million taels of silver(Cambridge History of China Volume 8 The Ming dynasty,1368-1644, Part 2 pg 298) but the Chinese invested more troops in the Bozhou rebellion and Ningxia mutiny.

Though hindered in Korea's mountainous terrain the Ming had plenty of cavalry that would see in action in flat terrain which the Japanese never encountered in such a large number,the Japanese also lacked artillery that would make breaching Shanhaiguan(if they even manage to reach there) impossible.

Though the Ming preformed quite dismal in later sieges,they were able to hold their ground against numerically superior Japanese forces in the Battles of Byeogkjegwan and Jiksan.

Not to mention the Ming forces were crippled by logistical issues,the Joseon dynasty even had trouble providing horse feed to the initial relief force of Zu Chengxun(帝國最後的榮耀:大明1592抗日援朝),while Song Yingchang sent letters containing scathing criticisms to the Ming court about the Joseon's lack of good faith and empty promises(經略復國要編).

Even so the Japanese had to contend with he 9 garrisons in the North had 651,665 troops of various quality while maintaining 279,158 horses.(邊鎮糧餉明代中後期的邊防經費與國家財政危機) as well as countless militias.

From the 神宗顯皇帝實錄 and 經略復國要編 the Ming were afraid of a Japanese landing in Shandong,therefore there were orders to establish outposts in coastal provinces and station troops.

However Hideyoshi succeeded in angering the Ming,Joseon and the Jurchens effectively ending any support to conquer Ming.

While the cost of the Imjin War was rather high it wasn't enough to cripple the Ming government,I would put more weight on the depreciation of the Ming fiscal policy as well as the mini Ice Age.