The trouble there is that the Emperor really helped to bind the country together. No matter what the actual political situation looked like there was always the emperor of Japan whose religious role is often gravely overlooked.
Or grossly exaggerated, IMHO. I've read a fair amount about the Sengoku period, and the Emperor was nothing.
Plus, of course, ITTL, Japanese have mostly become Catholics, for whom the Emperor's "religious role" is nothing.
Japan was traditionally seen as a true empire, a collection of different countries, not a mere kingdom with separate regions. You can see it in the names of several places in Japan, e.g. 四国-Shikoku-4 kingdoms.
One of the major goals of the Meiji Restoration was in destroying the ancient order of Japan as an empire of different lands and creating a single truly united modern nation-state.
I would see Japan fracturing before any single man establishes a new shogunate. The chaos of Catholicism overtaking the country would provoke a lot of fighting in itself.
This raises a possibility - Catholicized Japan may not remain united. This seems unlikely, as there was no history of a separate state in the Home Islands in the previous millenium, but
perhaps that was because of Imperial symbolism.
As to the existence of a non-white power- I really wouldn't see this as such an issue. Modern conceptions of race didn't really exist at all at the time.
Let's say non-European. And while "race" in the 19th century sense was not a concept in 1600, I think Europeans made a very sharp distinction between them and everyone else.