I was mainly thinking of a member of the Imperial House founding a Shogunate, like the Minamoto. Or maybe some sort of alternating system, with the Shogun's son becoming Emperor and the Emperor's son becoming Shogun. Like how the Northern and Southern Courts were supposed to alternate on the throne.
I also like the idea of a switch-up in the Tokugawa Shogunate, with the new Shogun being an Imperial Prince. What would the consequences be? Would the Shogun become a puppet ruler like the Kamakura Shoguns or would we see a member of the Imperial House wield actual political power? And if the later, would the Shogun try to claim the Imperial throne as well, creating a much difference and earlier Imperial Restoration? Or would any Princely Shogun have to renounce the Imperial Throne first?
Neither of your ideas are likely. Remember that the Tokugawa family was not extinct after Ietsuna's death, which is why his brother and heir Tsunayoshi was picked. Contrast this with the Minamoto of Yoritomo's line, which died out in the first or second generation after him, which gave the Hojo a reason to pick another family. Picking anybody else besides a Tokugawa would basically be against that family's law. Deviation from principle would be quashed by the many wealthy and powerful Tokugawa lords, including Tsunayoshi, who is a powerful lord in his own right. For example, he has a han with an income of 250,000 koku of rice, which is something like twice the income of the entire imperial court itself.
As for the princes who were Kamakura shoguns, they are all basically irrelevant while the Hojo are in power. Now, there was a restoration of imperial power under Emperor Go-Daigo, whose son was Shogun, but this was only when Go-Daigo was allied with several military lords. This alliance can only be sustained in opposition to the Hojo. Once this breaks down, I predict that one of the lords would try to become Shogun, as happened historically with the Ashikaga. The other alternative is for the Nitta to become Shoguns instead. Both of these families trace their descent to an emperor as well. Now, during the end of the Kamakura era, the power of eastern military lords is at its peak and the powers at the center, whether in Kamakura or Kyoto, are at their nadir. So it is unlikely that an imperial shogunate could form, when the capitals are dependent on provincial lords who have all of the money and the men. The interests of these lords were opposed to the centralizing tendency of Go-Daigo.
There might be a situation somewhere in the middle. Let's say the Hojo retain Kamakura and the east. Given how the Ashikata and Nitta have domains there, they'll remain in opposition to the Hojo whom they hate. And to balance out these two, maybe an imperial prince could be shogun to placate both sides. But once there's unification, military men will rise to the ascendancy. This is an overall trend that a few decades or years of imperial control will not break.