I agree with the consensus here that you can do a 2" strip for your boarder. Sewing only one side of the fabric to the front folding it over and sewing it down by hand to the back with a single fold under. It is doable.

There are more ways to maximize the fabric you have available for binding. One is Tartan's way. You can put that on upside up or upside down. If it is upside down, the "scrap" will be mostly showing and the "good" fabric will be a flange. So you would make the scrap side in fabric that you want to show.

More savers: Don't cut off your selvages. They will be mostly cut off as you make your seam on the diagonal. The tiny bit still on there is then in the seam allowance. I usually cut off the very ends that are very thick. Less than 1/8 of an inch.

Don't make 45º seams when joining strips. I lay my strip on the line on the mat with the end of the strip on a line and the strip edge on a line. Lay the another strip overlapping this just one inch. Use a ruler to cut across on the angle from the end of one strip to the end of the other one. The lines on the mat will show you where that is. Lift ends of strips right sites together and make the seam. The resulting slant is not 45º, but it's enough to keep bulk from the seam from piling up in one place.

I hope you can understand this. I do this even if there is plenty of fabric. I like to use leftovers in my scrap quilts.

Because it is a smaller top, the quilting should not need that much extra backing - just try to be careful with your quilting. If you only leave 1" on each side, that should free up enough for your binding. Another option is to do a self-binding. I often do this on a baby quilt because then I never have to worry about the baby chewing off the binding! Then just leave 2 1/2" on all sides, after quilting, trim to 1", fold over cut edge, fold over to front and topstitch. This gives you a 1/2" binding that will not come off. Because it is stitched thru, it gives the back a nice 'framed' edge.

Sorry - haven't read all of the posts thoroughly. I measure my quilt top - not the excess batting/backing to determine my binding requirements. I cut at 2.25" and if my measurements come up with x strips (wof) plus y number of inches, just cut an extra strip (WOF) and have not had issues. Good luck.

This method would require no binding. https://www.redpepperquilts.com/2011...-tutorial.html

Once when I had this problem I solved it by not cutting the binding until after quilting. I laid out the quilt as close to edge of the backing as I was comfortable, so that the excess was all on one end. After quilting I trimmed off the excess, and since that 2 inches of extra was still attached I was able to get that extra binding strip out of it.

I agree with the folks who talk about using the scraps from trimming the backing after you have quilted it. I often take the part I trim off and turn it into the binding. actually I like using it because most of the time, I don't have to join as many cross cut strips as the backing is longer ...when I long arm. I will leave the shelvage on the backings where they get pinned to my leaders...then I start quilting as clos to the leaders as I can get. So often I have quite a bit of fabric left that I can get a couple of strips of binding from.

Never thought of doing that before. I can see where it could save my bacon one of these days. I usually cut my binding 2" wide. I like it on the skinny side, but I'm also guilty of not planning ahead and sometimes run short.

Not sure how much other fabric you have left over from the top but you can do pieced binding. Like this one.
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Are you a glutton for punishment? There's no way I'd want to struggle with 2" for a binding. A baby quilt will be washed a lot, so it's important to have a generous seam allowance along the edge. I would probably buy some compatible fabric. As long as you get the color right, a small print or solid should work just fine. A stripe that echoes the colors in the quilt might be a cute option.