The two biggest non-self-inflicted challenges the Japanese had to deal with during the Imjin War were 1. the Korean navy and 2. insurgencies in occupied territories. Buddhist monks in mountain monasteries and yangban-led peasant 'Righteous Armies' constantly harassed Japan's already overstretched supply lines independently of the Joseon and Ming armies. Seeing how poorly received the Japanese were during the war by the populace, it's doubtful that a Japan in the throes of civil war between the Toyotomi and Tokugawa will be able to maintain control over Korea, especially considering the war would be within a generation of the Imjin War (if we're going with Hideyori losing control of Japan).
There's also the Jurchen/Manchu to account for, with Nurhaci having offered to help the Joseon expel the Japanese during the Imjin War and the Japanese attacking Jurchen territories during the war. The Manchu rolled over the Korean army OTL twice without any help and that's without the Koreans having to deal with insurgencies elsewhere. Plus, reinstating the Joseon dynasty would be easy, with how unstable the Japanese regime would be, and would score Nurhaci brownie points with the Joseon and the Ming, which he was going for when he offered to fight the Japanese during the war. It'd give him a valid excuse for grabbing supplies from Korea that would help with future wars with China, prevent the Chinese from being too hostile with him early on while he's still consolidating, and secures a neutral or allied southern border (which Hongtaiji would do later on after the more neutral/pro-Manchu King Gwanghaegun was overthrown in favour of the pro-Ming King Injo) that owes him a favour, rather than a regime that had attacked his lands without much reason very recently and would serve as a liability (since Joseon loyalists would side with the Ming if the Manchu don't support them instead, and that means committing troops to subduing Korea while fighting with the Ming and potentially Tokugawa, which isn't exactly a sustainable model long term).
Most of the Toyotomi power base would be lost if the Tokugawa take over the home isles, which leaves the Toyotomi and their allies in a precarious situation if they retreat to Korea, where they likely won't have much support amongst the locals and not enough time to build any sense of legitimacy or loyalty, seeing as they were invaders that brutalised the peninsula just a few decades (or less) ago.
This is compounded with the fact that the Tokugawa will not suffer the Toyotomi exist any longer, as the Toyotomi are a direct threat to their legitimacy as the hegemon of the Japanese islands (the tozama daimyo of OTL who sided with the Toyotomi were sidelined for over 200 years after the fact and ended up overthrowing the shogunate, after all). A resurgent Toyotomi clan would naturally try to reclaim their ancestral holdings and powerbase, especially with how much of the archipelago sided with them when push came to shove and still remained loyal/bitter against the Tokugawa, which is a threat the Tokugawa would have no reason to not try to kill in the cradle before it can actually manifest.
So the Toyotomi will lose all the lands they actually had effective control over, they'd be stuck with rebellious holdings that are liable to uprisings and invasions from the north, and the Tokugawa will try to end the Toyotomi at the first opportunity to remove the security risk.
I'd say it's not very plausible, unfortunately.