Depends how much less you have in the one corner. If it is too skimpy you might be causing yourself more frustration than you would if you just bound it with a different fabric and not do the self binding. I wouldn't attempt it if you only have 1/2" extra to fold over to encase the raw edge and bring around to the front, especially if you used a high loft batting. If you have 3/4" of an inch it is fussy but doable.

Edited to add, don't forget that the actual quilting process draws things in a bit as well. You don't mention if the quilt is already quilted or not. If it isn't quilted, then you can do as others suggested and simply sew on an extra bit of the same fabric if you still have some and if not, a different complimentary fabric to get it up to size.

or patch in a contrasting piece that you might consider an "accent" - make it obvious so it looks " intentional"

without seeing what you are doing, this suggestion is only semi serious.

I haven't read all the replies but here's something I've done and it looked great.
I took fabrics from the quilt and sewed them together to make a binding. It looks awesome!
Whatever you do good luck. I'm sure he will love it. You do such nice work.

I would finish it with traditional binding or even a flange binding. This is a perfect opportunity to play with binding techniques. That boo-boo would haunt me forever if I tried to patch it in. I know myself well. LOL!

Remember, you asked what we would do. Do whatever makes you happy with your quilt. Your DH is going to love it!

This is what I would do. I find self binding is more trouble than just making a separate binding.

If the backing fabric is a "must" for binding, could you cut widths for binding and add a coordinating strip to the center of the backing?

Just add some of the same fabric - I've done this and it's only noticeable if one knows it's there. Good luck!

It depends on what you want for your finished result. Self-binding never seems to look as neat as regular binding. I was going to do that and after the fuss of folding over the fabric and trying to get it to perfectly match on all sides, I decided to use regular binding. However, it may be best for your project.

Thanks to all who replied! I did end up sewing a bit onto the shorter side and it worked! Thanks again!