That's to get approval for a venture of the sort. Then comes the actual financing, recruitment, negotiations to not get involved in other European affairs that are more pressing (no Spanish Armada to begin with would've been more conducive to this plot, seeing as the Spanish Armada invading England would only be the first step; then comes the actual fighting in England, garrisoning English forts, suppressing rebels who have French and Dutch aid). A successful invasion, that's a whole other matter.
Anyways, there's no promising the Chinese would even need Protestant support to kick out the overextended Spanish to begin with. And even if they did, why would they brook Christian conversions? That would be seen as part of what caused the mess to begin with, plus there's the whole matter of if Christian beliefs can be reconciled with Chinese ancestor veneration and all (Martin Luther did . It's not impossible, but it's not something to handwave.
Well, there was Yi Eokgi, albeit a bit young. Won Gyun wasn't brilliant, but he probably wouldn't have seen his promotions without Admiral Yi's successes, so who becomes the supreme naval commander so not something I can guess. In any case, the Joseon fleet also wasn't virtually wiped out until the 2nd phase of the war, after initial peace negotiations, and by then the Ming had already established enough of a military presence in Korea that the Japanese failed to make much headway (they managed to make it a quarter of the peninsula that time). So I wouldn't say the Japanese supply lines would be undisturbed. Much less damaged, sure, but the Japanese naval commanders were focused on looting over fighting at the outbreak. The Joseon fleet probably wouldn't have seen the same level of success as OTL without Admiral Yi, but it wouldn't have been a complete nonfactor.
And again, Japan can't stay in Korea more than a couple of years, really. Their own internal politics and Ming armies actually reaching the scene (the Ming had only between 17k and 50k troops in Korea in the first phase of the war, when they pushed the Japanese back to the central region of the peninsula, then increasing to 98k in the second phase) would've ended that in short order. It's worth noting the Ming during negotiations in the ceasefire thought the Japanese were effectively surrendering all their gains. Well, until negotiations broke down once everyone realized the others were on different pages altogether. In any case, once 1616 rolls around and Nurhaci has unified the Jurchen tribes, then there's another factor to consider (the Manchu Invasions were triggered by political events in the Joseon Kingdom and often involved Joseon defectors fleeing political strife and seeking revenge. The Manchu in turn obliged because they wanted a friendly Korea to their southern flank). Rather than trying to bail out hyperaggressive Japanese daimyos busy with fighting each other, Tokyo, and their own subjects, installing their own puppet king over the peninsula sounds much more Nurhaci's style. But that requires saying the Japanese actually hold on until then.
Back to a less handwavy view of the war, the main detriment to Korea in this scenario is, if the Ming do the heavy lifting in liberating the peninsula, the Joseon are now even more obliged to tie themselves to the sinking ship that is the Ming dynasty due to Confucian norms. That was what triggered the Injo Coup and the following Manchu Invasions, which crippled the country for decades.
Culturally, the main thing the invasions caused were a calcification of culture, growing isolationism, and even stricter moral restrictions on women. The royal court felt shaky after the invasions and needed to shut out dangerous outside influences while the widespread rape of Korean women by the armies rampaging through the peninsula led to a weird hyperfixation on the ideal, pure Confucian woman, untouched by war. This stuff is unlikely to change in Yi-less scenario and perhaps might become even more entrenched. OR the Manchu install a puppet, in which case...I don't know. Crown Prince Sohyeon, a Manchu hostage, was apparently quite interested in Western medicine and Christianity after encountering a Jesuit priest in Qing Beijing, but that was his quirk and the general response was not...positive (his father King Injo allegedly beat Sohyeon to death with Sohyeon's present to him or poisoned him soon after he returned and talk about western medical and religious practices. Who knows, Injo was a terrible king).
And I'm not counting the case in which the House of Yi is wiped out because the Joseon were open to tracing back the family line as far back as possible to select the closest agnatic successor. The Joseon once picked an illiterate peasant who was a great, great grandson of a king a century before to be the new King after the previous line died out. As for why he was an illiterate peasant, the Joseon royal family was pretty fond of poison and exile, so there were royal descendants roaming around, subsisting in absolute squalor, far from the capital. Which made it harder to wipe them all out.