I agree with waxing/conditioning most hand quilting thread but not sure how that would work with the pearle cotton. To me, it would almost defeat the purpose of that particular thread/style of quilting.
If you haven't gotten too far into the quilt, definitely try a different thread/needle and see if the bearding continues or not.
Your other option is to continue on your current path. Since you mention this is a quilt for your youngest who wouldn't know/care about the stitching, just continue with your current plan and finish as you had planned. After washing, the bearding probably will not show at all.