I too have made several of these with no distortion. I used a flat batting (Warm and Natural that I wet and dryer dried to preshrink it a bit) and I outlined the major figures and did a filler stitch around the major figures. In the case of the adorable baby, I would outline him (either by hand or machine).
If it was me, I would outline his limbs so he is emphasized. I have actually hand embroidered through the pictures and the batting (not the backing), then added the backing and filled in the background with stippling and cross hatches. I loved the look -- kind of a fake trapunto.
I have only use Printed Treasures fabric to print on and I have used abd HP printer. The fabric is a bit more dense than regular quilting fabric, but it looks good when quilted.