I haven't done a t-shirt quilt, but I would use the lightest weight fusible tricot I could find. I have used both the tricot sold at JoAnn's and a tricot I got from a drapery company online on silk ties. The fusible tricot from JoAnn's was expensive for a small amount; the fusible tricot from the drapery company was inexpensive (about $2 a yard, I think, plus shipping). Also the drapery fusible tricot was slightly lighter in weight than the package I got from JoAnn's. The drapery stuff was sold as yardage and I think was 62 inches wide, so you really got a lot for your money.
Fusible tricot is fairly lightweight and has some stretch; that's why I would choose it for t-shirt fabric.
It's a good idea to steam fusible tricot for a few seconds before fusing to allow it to shrink.
If you try something else, it would be a good idea to fuse a scrap piece of t-shirt first to make sure that you are satisfied with the stiffness and drape. Many fusibles will make the fabric uncomfortably stiff. It's also a good idea to test wash the fused piece. The wrong kind of fusible may make the fabric bubble or wrinkle when you wash it.
Most people just stitch in the ditch for a t-shirt quilt.