One of my grandsons dragged around a satin pillowcase - all over - until he was about 3. that was about 20 x 32.
I think a "drag-around" quilt could be fairly small. Then when child outgrows it, you have a reason to make another one. :-)
I think I would tie it if I was going to put fleece on the back.
Maybe the only caution would be for everything to be equally washable.
I've used brushed arnel on the back, but that has very little stretch.