Oh boy now that's more like it!
I have heard the stories about the "daito mansion." The question of its truth is of course the main issue with that... but besides that, it confuses the yamato-vs-daito issue even more. The pronounciation of 大東 within that "legend" is presumably straightforward. If Sokaku Takeda had exposure to that purported history, there is double reason that he would have been saying "daito" instead of "yamato."
(Reason 1 is "yamato" has its own kanji unrelated to 大東, reason 2 would be that 大東 refers to some kind of true history that people knew about and were talking about).
So instead, we are left with a legend that was made up/tied to the art after Takeda started teaching, and again, a strange kanji mixup.
Maybe in the absence of any real history, there was no name and no kanji at all for the art, and he was just calling it Yamato ryu. Later there came along some good reason(s) to just shift the name to Daito.