Well, from what I've read, they didnt turn them down, they just demanded too many changes. After Sega turned them down first, PlatinumGames said that in the many attempts to get funding, people either said No, or would request changes. They would ask for certain content to be cut or changed, to have content pertaining to their franchise's added to the game, all while threatening to pull out if its not going how they wanted.
Nintendo offered to fund Bayo2 stating that they would give PG as much money as they needed in order to make their vision for the game to become a reality. Nothing was asked to be cut, changed, or exchanged with something else. The whole idea of Nintendo based Costumes that appeared in the games were Hideki Kamiya's idea, with a star Fox Costume sketch. and then Nintendo suggested other costumes to expand on the idea.
Did you know that initially the Link costume featured a buttoned up chest because the programmers didn't feel comfortable sexualizing links clothes too much? But when shown to Nintendo executives, Nintendo said "undo the buttons on her shirt, having them buttoned up like that doesn't feels like it doesn't fit her character." So thats why she has the open chest with the Link costume.
Other companies were being greedy and forceful with the money they were practically blackmailing PG with in order to fund the game, where as Nintendo gave them everything they needed and let them make the game they wanted.
And what I don't understand is how people can still 'expect' an 'eventual' port of Bayo2 to other consoles. Seariously, you mention Bayonetta 2 on sites like siliconera or IGN, or even other gaming forums like Neogaf or KoeiWarriors, and you will STILL see people claiming that its going to happen. Stating things like "its Sega's IP, they can move it to another console if they want." and some how fail to realize that Nintendo co-owns the second game due to the fact they funded it. They try to come up with ways to validate Sega breaking their contract with Nintendo in order to make more money off Bayo2's sales on other consoles, and are completely ignorant to the heavy series of lawsuits that would follow for breaking said contract.
Like there's this one guy on the KoeiWarriors Forums who, every time someone updates the Bayonetta topic in the Action Games sub section, gets upset that it wasn't news about a re-release of Bayo2 on another console. They were even stupid enough to think that the upcoming Bayo2 re-release, for Bayonetta getting into SSB4, was news about her game getting on the PS4. I tell you there's ignorance, and then there is imbecilic. And this guy is waaaaayyy deep in the latter of the two.