Since you're asking about stabilizer, I'm assuming you're going to fuse this as an applique? Or were you going to do some sort of embroidery?
This is how I would do it if it were my quilt. I wouldn't use Kona, I agree with you about Kona's hand. I'd find some other quilt shop-quality black fabric (I'd use something with a subtle print, but if you want a solid black I've heard good things about Michael Miller's black fabric). I would not use any kind of fusible. If you're going to machine-applique it on, use a zigzag or buttonhole stitch with black thread around the edge of the silhouette, then cut away the background fabric behind the applique. Quilt as normal. The zigzag or buttonhole stitch will secure the quilt, and you could quilt a cute design in the silhouette to make it even more secure. I don't think you'll have any problem with laundering.