There is a danger in picking something that requires too much research to be done if you want to be historically accurate, I think. Personally I'd recommend taking those elements that interest you and then redoing them in a "Japanese past," if you were going to do this story. (So, using something like Go-Toba's exile to the Oki Islands, but not naming him Go-Toba. You have the geography in your head, so references to that shouldn't be a problem. But then, if you had something like daimyo or hostage systems or anything else that didn't work for the actual time period, you wouldn't be flagging it so much.)