I don't know these buttons, but I would think of how she would be buttoning the sweater as she wears it. The buttonhole won't have to be the 1 1/4 dimension, just wide enough for the 9/16 if she can slide the button into the hole sideways and if the button is wider than it is tall.

I fiddle with casting off 2 or 3 stitches in one row, then in the next row, casting on one more than I cast off. The third row, I knit the extra stitch together with the first stitch after the buttonhole (or before) to tighten it up a little so it is not too gaping.