I think your idea of layering a plain piece of fabric with the batting would work....or maybe you could just "line" the leftover bits and not the whole back? I like to use white too, but I have never had the back shadow to the front. Maybe you need to change batting? Warm and Natural is what I use, in case that helps.
Or maybe two layers of batting...fluffier and wouldn't show thorough?? Just sending out ideas here.
Dina