Which Sengoku-Era Clan would you want to rule Japan?

[Paradoxer]

The Emperor himself wasn't but the court still had plenty of power during the Kamakura shogunate, at the time Japan was in a "diarchy" of sorts where the Imperial court ruled western Japan and the Shogunate the east (mostly Kanto and Tohoku) at the onset of this system the Court was arguably the senior diarch, being able to freely appoint a Shogun and greater saying in government of the provinces, this would change following the Jokyu war of 1221 where Retired Emperor Go-Toba tried to destroy the bakufu, this didn't work out as intended because Go-Toba wasn't the most popular guy in town and the bakufu managed to raly even non-Shogunate samurai to their cause, in the aftermath the bakufu managed to have a say in Imperial succession by sending a secretary to Kyoto and managed to make appointments of military officials in Western Japan, but the court still had rights and authority there, they just bickered heads whatever possible though.
Not sure how to restore this system after the Sengoku, it collapsed rather unintentionally as the Court relinquished fealty of Western (or just Kyushu, sources aren't exact) samurai to the Shogunate for the defense against the Mongols, which caused the shogunate to collapse half a century later and the brief restoration of the Imperial family, Emperor Go-Daigo's policies alienated one of his chief supporters Ashikaga Takauji that rebelled and established the Ashikaga (or Muromachi) Shogunate, Go-Daigo and his descendants would continue to fight Takauji and the Ashikagas for the next 60 years in the Northern-Southern Courts period (an earlier Sengoku), during this time both courts would sell off land and administrative privileges and rights to prominent nobles and samurai to win their financial and military support, leading to the rise of the shugo(-daimyo) and complete feudal-militarization of Japan, the Ashikaga eventually prevailed under a compromise that the Northern and Southern lines would alternate between the thrones, but immediately broke their promise and culminated in total submission of the court to the bakufu. So by the 1500s you had: Japan completely dominated by military feudal lords and the court was completely powerless as the bakufu has finally dismantled the court military capabilities and landless as marauding warlords had seized imperial and kuge estates following the collapse of Ashikaga authority, the Onin War that devastated Kyoto broke it even further, prominent courtiers went to the countryside to make a living by tutoring young samurai and the Imperial family himself was in poverty, having to sell calligraphy on the streets of Kyoto and wasn't even able to enthrone a proper Empress for 100 years! Commentators even note the "Imperial Palace" was barely distinguishable from regular housing.
Nobunaga and Hideyoshi surely tried to restore the fortunes of the Emperor and court, but be aware they didn't had Imperial Restoration in mind, that wouldn't benefit either of them and honestly neither had the authority nor means to re-institute direct court rule, rather their objective was to use the court as a mean of legitimacy, as restorers of order, having the court share power would just undermine their own objectives as although several nobles and princes were grateful and even supported them in person, the old kuge-bushi dichotomy was pretty much alive and the nobles would rather not equate themselves with brutish barbarians (Hideyoshi assuming the office of Regent was seem as pretty scandalous).
The Imperial Restoration of the 19th century was something truly special, but because it was possible due the introduction of Western ideas of rule, a monarch like the Emperor fit the mold, a technically appointed military dictator like the Shogun didn't.

I’m more aiming at shogun intermarriage with Emperor bloodline tying two more together. One “Shogun-Emperor” before splitting into different family branches but still one. Maybe make shogun male succession and Imperial female(empress) to possibly help tensions. Not bringing back courts but Emperor/Empress becoming officially “absolutist” but in reality power is limited and passive to shogun majority of time.

The shogun gets the Daimyo in line and make them have clear cut succession and territory. If they want more they are allowed to fund their own expeditions across pacific or foreign lands. Sometimes with shogun support or aid(get sons and samurai who lack inheritance to leave and hopefully expand realm in process. Similar treatment towards rebellious people).

The monarch is one that deals with public, non-samurai/nobles, national economy, scholarship/academics, diplomacy(they go talk and deal with foreign leaders), mediate issues with daimyos and shoguns, various economics the shogun doesn’t so civilian economy, navy so they have some actual force or leverage somewhat, foreign affairs, and religious affairs. Or anything shogun doesn’t want to deal with.

Shogun handles anything related military, law enforcement, security, and daimyos. The monarch is given imperial cities and some major urban centers or his extended family. Daimyos control much of other land.

The Japanese do need to expand or they will have repeat of domestic conflicts between large samurai class, daimyos, and shoguns. A bunch of Ronan is not good for home islands but better used overseas. Maybe emperor even hires Ronan to support navy and for anything relating to merchant navy or colonies(“imperial colonies” which are small islands, major trade post or ports, or growing urban centers).

The shogunate and daimyos basically do away with imperial courts with shogunate taking its place directly. Basically, daimyos and samurai assembly where lords appoint representatives on their behalf in capital maybe second or third son. The Shogun and Emperor are also made to share same capital and residence next to each other. Making plotting and civil war harder. At worse coups might happen but usually quick and much much less destructive. The emperor also has favor and support of merchants, fishers, artisans, navy, traders, scholars, monks, and urban populations.