You want that stabilizer to stay in the quilt. Those knit t-shirts will stretch between your quilting lines and you'll end up with stretched out spots. What you're trying to do with the stabilizer is make stretchy fabric behave as if it were like the cotton fabric we use in our regular quilts.
I just buy the cheapest, light weight WHATEVER fusable interfacing at JoAnn's when I have a coupon. We're talking about $.50 per yard. I bought a bolt the last time I did this because I also use the same stuff to fuse together pieces of batting for charity quilts. For less than $10 you will end up with a pretty big t-shirt quilt with enough scraps to piece your batting scraps.