Do you know if this binding fabric is woven or not? If not, you should find some binding made of bias cut woven acrylic like Sunbrella. The bias cutting may not help much for the tight corners, but it will surely help the edge over your door window. Also, a binding attachment for your sewing machine is not a bad idea either. Sailrite has all the Sunbrella bindings. Give them a call, maybe they have other ideas for you.
No one lives forever, the trick is creating something that will.