You arent the only one to wonder about this - been there, done that!
I asked around to get some advice on this one and a friend told me that when she binds by machine that she sews the binding on the backside and flips it to the front so that you are machine finishing the binding on the front side. You dont need to change the width of your binding. You are making the front look nice and neat because you can actually see what you are doing. You will see those stitches on the back so just load the bobbin with thread that matches the back.
It wont be perfect (mine wasnt) but it looked much better than it would have if i tried to work blindly hoping i would somehow catch all the edges. It was always a big disappointment to me because i DIDNT. This works.