You can use different type of materials and fibers in quilts. Sports stuff is designed for use and washing and often holds up very well.
I'd be concerned about how good a bond you could get with an iron-on fusible without harming the actual shirt. I think though, a light weight muslin or batiste underneath would be a good way to deal with the more mesh or the poly types with all the breathing holes. Takes a bit of care dealing with slipper fabrics, I might cut the pieces with a half-inch seam -- doing a good zig zag in that extra seam allowance. Depending on size, some larger ones you might even need to baste or safety pin together until some quilting down is done.