Starch the fabric heavily. don't stretch it when you hoop it and float another piece of stabilizer under the hoop.
My personal thought is that this design may be too dense for quilting cotton. You can probably get it to stitch out without puckers but it will be like having a piece of cement in the middle of your fabric. Less dense designs work best on lighter weight fabric, denser designs work well on heavier ones. In one of the first classes I had with my embroidery machine, the instructor told us that selecting a design with appropriate density for the project was one of the most important things to learn.