It's a great quilt! I inherited a scrappy lemoyne star quilt top from my Mom, who inherited it from her Mom 40 years ago. My Grandmother probably made it around 1940, so it was old, too. The green in the quilt looks like the same color, and it was the background fabric, so there was a lot of it. It was a challenge finding a fabric that worked well with that green for backing and binding. I finally found a pastel green/blue/off-white plaid that worked perfectly.
I'm "color challenged" so my advice on whether the green flowers will work or not is pretty much useless. (My DH had final say on the binding for my Grandmothers quilt.) I think I would ask if there is any more of the backing fabric or the green border fabric available, and, if so, use one of them for the binding.