I prefer the buttonhole stitch. You can adjust it to be small enough and close enough together that it shouldn't cause a problem when laundered often, especially if it's fused.

A friend of mine, who machine-appliques often, uses the blanket stitch set to a tiny length and with. It gives the applique a very nice look.

Someone earlier mentioned using the zig-zag set to a tiny length/width. I saw in a machine quilting recently that an award winning quilter had a very complex quilt top with tiny geometric applique. She attached the applique with a very tiny zig-zag. After seeing her do that I used it on a quilt of mine and was very pleased with the result.

Hope these comments help you decide.

I did a whole quilt of appliquéd dogs. I did buttonhole stitch with black thread. First matched the thread to the dog colors, but decided I liked the "pop" of the black. Might not work on your soft colors.

That design begs for the buttonhole stitch.

Cute quilt. Although I do buttonhole myself, I know a quilter who uses zig-zag, but always uses buttonhole for clothes. She would zig-zag the dog and buttonhole the coat.

I agree with most of the other posts. I would use a small buttonhole stitch. I don't like the satin stitch for appliqué, it is too stiff, and draws up the edges. Your design is so cute--perfect for a baby quilt.
Sue