Hi Megan! That sounds so frustrating. I'm guessing the manufacturer did not foresee the multiple layer issue so no, I don't think you have to do what they told you to have your project turn out.
What about trimming the wrap n zap so there's only one layer in the edge where you're sewing through so many layers? Or if you don't want to backtrack, maybe you can handsew those spots? Leatherworking tools could help if it's really fighting back.
Good luck! Let us know how it turns out.