I rambled a bit, so here's the short answers; Fictionalised historical characters are a-okay and the names are not especially troublesome, do what you think is best.
It's not going to make any particular problems, no. After all, they're not the game universe's version of the historical figures they're fictionalisations of, even if depending on how fictionalised they are it might look like it for a while. In fact it might be cool to have that little red herring floating around.
On the names! I'm going to leave that to your individual discretion based on this question; 'How plausible would it be to have that name?' For example I went to school with a James Cook, a pretty famous guy but also a name that isn't a stretch in modern day. But Isaac Newton would turn a head or two, because both are uncommon in modern day and who with the name Newton would call their kid Isaac anyway? I understand it might be a bit harder to decide this sort of thing with the Hakuouki cast, because it might take some research.
There's also always the option to have them be specifically named after their historical counterpart. It doesn't actually make any less sense than characters from entirely different realities having the same name. Might need to look up if that's considered poor taste in the culture though, I know it's thought of as a huge faux pas in China and some other places. And it's pretty unlikely they'll be named after an infamous figure from history unless you want to make a thing about it.
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It's not going to make any particular problems, no. After all, they're not the game universe's version of the historical figures they're fictionalisations of, even if depending on how fictionalised they are it might look like it for a while. In fact it might be cool to have that little red herring floating around.
On the names! I'm going to leave that to your individual discretion based on this question; 'How plausible would it be to have that name?' For example I went to school with a James Cook, a pretty famous guy but also a name that isn't a stretch in modern day. But Isaac Newton would turn a head or two, because both are uncommon in modern day and who with the name Newton would call their kid Isaac anyway? I understand it might be a bit harder to decide this sort of thing with the Hakuouki cast, because it might take some research.
There's also always the option to have them be specifically named after their historical counterpart. It doesn't actually make any less sense than characters from entirely different realities having the same name. Might need to look up if that's considered poor taste in the culture though, I know it's thought of as a huge faux pas in China and some other places. And it's pretty unlikely they'll be named after an infamous figure from history unless you want to make a thing about it.