Shibori - the fusible web problem may be due to the brand (manufacturer or company) of batiks you have purchased. The major brands don't have this problem in my experience - I make bags/totes frequently and use fusible fleece with the batiks often instead of batting for structure.
The batiks with a more limp and lighter weight feel to them seem to be made without good standards even though they may still be priced in the high range! Although their patterns may be appealing, I choose not to purchase them. Hope this helps you not give up on batiks!