I think you will need bias. Trying to use straight grain on curves starts out fine but it start to make a ripple in the binding and it kind of rolls until the binding gets all wonky.
I have used straight of grain binding on a straight quilt edge and switched to about 10 inches of bias binding on all 4 corners for rounded corners. The binding was dark so it didn't show the seams where I attached the straight binding to the bias at each corner.