Congratulations on your first quilt. I have used fleece on the backs of quite a few Linus quilts. If I were doing it I would not even bother with batting. I would also make the quilt envelope style, like a pillowcase leaving one end or at least a whole to turn it. Fleece does stretch so pin the heck out of it before you do any stitching. This would save you the bother of binding. When I make the Linus quilts I use a decorative stitch to sew in the ditch or in some simple pattern that complements the top and then sew about 1/2 an inch inside the outside edges which gives it a bound look.
Depending on how young the child is they may be dragging it around for a long time to come. The lighter weight would give them a longer snuggling season.