Can you cut a very large over-sized piece of the fabric and still retain the element that you want to include? Even with paper piecing, when adding your next piece of fabric it is still going to be right sides together. You just need to make sure there is enough fabric. I usually place my piece of fabric (RS together) behind the paper, then pin on the stitching line and flip the paper back to make sure I a) have enough fabric where it needs to be; b) have the proper element where I want it.
Hope that helps. It's a really neat pattern, btw!