Very nice find. Well worth trying for a fix.

oldswimmer, the diamonds always have one side on the biase and the other straight grain. Therefore, they are easy to get out of shape. Once two sections are sewn together, that corner needs to be at 90 degrees, if it isn't sometimes you can square it up with a ruler and rotary cutter. After the four sections are square at the middle, you can resew as the others have suggested.
It will take some work, but should get a lot of the fullness out and maybe your quilter can do the rest when she stretches on the frame to quilt.

What a beautiful find :D:D:D
I wonder if you sprayed the center with water, if that would help the bias edges to shrink back as they are drying, if they were stretched during folding/handling? I wouldn't press it too much, as it could set the stains in it.

Others may not agree with me, but I would iron slightly and if it were still puffy, I would applique something on top of that spot like it was planned that way.

OK, here is a question from a complete noob. Is it necessary to remove the stitches from the middle and resew? Couldn't you just "squeeze" the puckers together and add another set of stitches?

WOW!!! What a great $4 find!