OK, so you're really looking for that bit of 3D texture from an actual weave.
Some random babbling thoughts....
The woven block will be two layers of fabric, foundation, plus the fusible. (I've done Anna Faustino's method from Woven Quilts so that's what I'm basing this on). The samples I have from here method is pretty stiff and thick, and I wonder if that might take away from the "softness" of your quilt.
That said, I just had a flash of solving your green triangle problem. Give me a minute to play with some fabric to see if it would work.