I made the entire "Planetary Party" series (designed by Amy Bradley) and it was so much fun! I'm a "serious amateur" astronomer and I collected really funky fabrics for about 2 years. Because I divided the quilt in half for two wall hangings, rather than for a bed quilt, I used fusible fleece on each block's background before I appliqued. This would make it too stiff for a bed quilt that would be washed but it was great for the wall hanging. The "monsters" were fused onto that backing and I used a blanket stitch for the outlining. I must admit that I haven't mastered the satin stitch - yet.
I use stabilizer that I can leave in -- tearing it off seems to distort the fabric when I do it. Same with paper piecing -- I use the stuff that stays in.
Do you think that you could carefully cut the monster away from the backing (leaving the stitching) and then fuse the whole thing onto another piece of stabilized backing? You might need to use something like "fray check" on the satin stitching before you try it.