I can't add any new suggestions to what has already been posted but just wanted to say congratulations on finishing the quilt and what a real bother to have this problem to contend with. I'm sure you'll end up just fine and I look forward to seeing pictures of the completed quilt! I'm like you, it takes me an age to get a quilt done, I feel quite like a tortoise when I read how much others seem to accomplish on this forum!

I came up short just a bit once, but when I put on the binding it nearly covered it. What did not get covered by the binding, I put the label over that section and "shhhhhh" nobody needs to know. It will be fine.

I don't have an EZ3 Grace frame, but do have the Z44. My instructions call for at least 4" extra backing rather than 2". I also run a basting stitch on both sides through the sandwich (an inch to 1/2 inch from the edge) as far as I can reach as I go each time I turn. I think this helps prevent slippage. I know this information comes too late for your current quilt. Whether you add more to the backing, use a wider binding, or trim 1/4", it will not matter. It will be a beautiful quilt you can be proud of. Congratulations on completing a hand quilted quilt. It takes patience and perseverance. I am sorry you are having problems with the EZ3. I love my Z44. I am just finishing up my third quilt on it with two more waiting their turn.

make your binding wider, and then cover it up when you bind it.

I would sew on another strip of the backing material to make it long enough, and then finish.

If it didn't matter that the quilt would be approximately 1/2 inch smaller and assuming you have a border around your quilt, I would just trim the quilt down. I would trim all borders to the same size.

Sorry about your backing problem, this happened to me the first time I used my Grace EZ3 frame. Since my shortage was a bit more than a 1/4 inch, I just added fabric to the back and then trimed, added the binding and no one was the wiser. I've had my Grace EZ3 frame for over 3 years now and have learned to make sure I have at least 4"- 5" extra all around the top.

If the border width not crucial I would trim 1/4" off all round. A wider border may not go far enough to overlap the shorter side.

[h=1]Extending a quilt back while on the Longarm: https://www.youtube.com/watch?v=Qs5Dw2raqfQ[/h]This may help for future knowledge.

I agree with this.
Making the binding wider will not solve the problem. You need to get the backing as long as it needs to be. If you have any of the backing fabric left (how about a little cut off the sides?), sew that onto the back where you need it. If you have none, I would sew a strip of the binding fabric there as a place holder. You only need an inch. It will be on the back, and will look like the binding is wider at that end, but will not detract from the rest of the back.

You might need to undo a few quilting stitches, and redo them after the piece is sewn on, but you will feel better about it the rest of your life if you do it RIGHT.