You will not need a stretch stitch if you are going to use batting and backing and quilt the blocks. Stretch stitch was used so that the garment would not tear when put on or taken off and during wear- it provided extra "give". Many polyester knit garments did not need zippers, but did need the extra give to fit properly.
Use a normal straight stitch and a new ballpoint needle, poly/cotton thread. You should be fine (except for the feel of that fabric!)