Sounds like a French Cleat Picture Hanger. If that is the case, and with only 1/8 of an inch to adjust, maybe you could lay a couple toothpicks horizontally on one end of the cleat that is attached to the wall.

Yea, I wouldn't use the same hole. Drywall has a bit more integrity than a patching compound. This is less important if you are screwing into stud though.

Graywings123: I had never heard the term "French Cleat," but I looked it up, and that's exactly what it is. The composite grips hold it off from the wall slightly, so rather than toothpicks, I used a piece of cardboard that I bent at 90 degrees. That way the part of the cardboard that went between the cleat and the wall was immobile, and the bend covered the end of the cleat just enough to level the Meural out. The toothpicks probably wouldn't have worked because of the gap, and because the Meural can be hung horizontally or vertically and gets changed every so often. But your suggestion was what made me change my entire approach to the problem and I sincerely want to thank you for your help.

Excellent! So much better than trying to re-do the cleat.