Does your fleece ravel at all? If there is little or no raveling, could you sew the pillow with wrong sides together instead of the envelope method? Then finish the seams by zigzagging them together. You would never be sewing through more than 2 layers at a time. It may not look as nice, but the pillow would still be comfortable to the animals, and with a long nap perhaps the zigzag stitches wouldn't be very visible.
Dayle