With or without stabilizer a T-shirt quilt IS heavy.
Regarding stiff, well it totally depends what kind of stabilizer you use. I have used the .99 cent a yard super el cheapo stuff JoAnn's sells and it was OK. I prefer the Pellon lightweight fusible for knits. I'm sure starch, if you use enough of it or a not so diluted mix if you mix your own would also work quite well. But no matter what you will have a very heavy quilt.