What an adorable quilt! I do lots of machine applique, most often I use the blanket stitch...and I rarely use a stabilizer behind.. unless 1 (or both) of the fabrics happens to be extra thin. Through the years, I've had no problems.
Here's a pic of a vehicle theme quilt I made about 5 years ago, I didn't use any stabilizer.. per the mom of the recipient it's been through several washings, and so far so good.
[ATTACH=CONFIG]570035[/ATTACH]
Regards,
Kif