Take some t-shirt fabric and try a 1:1 solution of Sta-Flo liquid laundry starch and water on it. (I paint this on with a large wall-painting brush, toss in the dryer, then iron with steam.) You will be able to tell if it stiffens the t-shirt material enough. I think it will, but I haven't actually done it with t-shirt material so don't know for sure.
Another option is to use MistyFuse. Although this is a fusible, it is *extremely* light and will not add stiffness. You would need a Teflon or applique pressing sheet to do this, as it does not have a paper backing. Amazon has some reviews of MistyFuse.
I have used tricot fusible for some applications. It does add less stiffness than other fusibles, but also adds some weight and thickness to the fabric.
I would perhaps experiment on old t-shirts (maybe from the thrift shop?) with these techniques to find the best one.