Wonderful, wonderful quilt!!!
I think what I would do is use my walking foot to outline stitch around each cat. It would be a slow process around the curves since you have to stop every few stitches to lift the presser foot to readjust the position, but it can be done. I like the walking foot because it keeps the stitches an even length for me.
I personally don't do SITD anymore because it wrecks my nerves and I am never happy with the result because every little deviation shows up. For this quilt, I would consider using a zigzag stitch to make straight lines over all the sashing seams (that is, go from one side to the other doing the zigzag over the seam).
My next step would be to do a medium meander on the black around each cat. I also might do a medium meander inside the sashings. However, instead of that, I might use the walking foot again to make rows of gently waving straight lines from side-to-side and top-to-bottom through the sashings. This would create cross-hatching in the cornerstones, which would be nice. As usual, I am having trouble finding a good photo online of what I mean. If you want, though, I will look more later and see if I can find a good example of the wavy lines. It's easy to do them with a walking foot, and they are not meant to be perfectly parallel; each line can wave a little differently. This means there are no slight deviations to catch the eye.
My preference would probably be for a colorful variegated thread throughout.