Lovely project! However you finish it will be wonderful.
I'd use a blue background border something like the top left star background -- something that has a bit of the tan and burgundy already in it. At least the width of the double sashed border, maybe as large as the center of the stars. I'd finish with the burgundy binding.
When in doubt/if you don't have the right fabric, consider a white or white on white binding. It would go with the whites in the stars and few things say nice crisp finish better than a white binding. If that's too stark maybe look at an off-white that isn't one of the beiges already used.