I've quilted down two crib sized quilts on a long arm using a minkee-like backing. My friend had put the minkee on the frame and then let it sit for several months -- it stretches! I was able to use it but had to take out what she had done previously and redo withe orientation on the frame turned around.
Had zero problems, would do again. My friend in Phoenix sends her minkee projects to be quilted, her quilter does not charge extra. She has a long arm frame with a medium throat machine -- she cannot do above about a crib size with her machine, just gets too thick wrapped around. She tried doing it halfway and taking it off and resetting it but she always got more of a lump in the middle than she liked.
So for the snowman applique, the big question is do you have a long arm? Or at the very least a deep throat sit down domestic?