You seem to be exhausting all avenues, so I can only make a couple more suggestions.
A needle with the largest eye possible. Wash the batik well in hot water, hopefully to soften. Turn the thread spool/cone upside down, it may remove the twist. If all else fails, try a good dose of silicone spray on the spool/cone. It will lubricate the thread, and will not stain the fabric. I have done this several times both on poly and cotton thread and it's worked miracles. I get mine at the hardward store, cheaper than the LQS (and some don't stock it). If that seems unbelievable, try it on scrap pieces first.
Love to know how you solve this problem.