Well, yeah, it's a site in Japanese with specifically Japanese subject matter, why is that a problem?
> I will leave cultural commentary aside. While the romanized names are large, they are incredibly awkward to read being rendered vertically with rotation
IMO, vertically-with-rotation is fine, probably even ideal, for reading the romanized text given the rest of the design, which presumably is appropriate for the main target audience.
The only real readability problem I see is that the text color doesn't change for contrast with the background color.
> I would recommend rendering the romanji vertically without rotation
Please don't. Consistently rotated context is easier to read as words than vertical words of horizontal letters.
> or using a Western left-to-right rendering
That's done for the Romanized text where space permits (e.g., for the selected color in the right-side banner.) But unless you are suggesting that the site should be designed first and foremost for Westerners, I don't see any more of that as reasonable.