Was there a large battle between the military government and the court? And what exactly was the Court in Kyoto like during the Shogunate? (for example, did they still dress in Heian court dress like the Junihitoe?)
There's actually been more than one shogunate in Japanese history. The political power shift from Court to shogunate happened long before the Edo Period, so I don't feel qualified to talk about that. However, I recommend reading this answer by /u/shakespeare-gurl on the history of the shogun-emperor relationship.
Formal Heian-style court clothing did persist for centuries, well into the nineteenth century, and of course is still in use by the Imperial family for formal occasions. (The current crown princess was married in a Junihitoe.) Fashion is not a strong point of mine, so I unfortunately can't elaborate or speculate on how the court clothing might have changed over time in minor ways, only that the clothing did indeed hark back to the ancient court.