Derek,
I am planning on CNC cutting / designing my own box, but I understand your question. The answer depends on whether the circumference of the bottom is the same as the top. If the box is tapered andngets a lot smaller at the top, darts only at the top might not be enough. The answer is to just try to wrap the cloth as you state with push pins to hold it up. Even if some conical shape changes the weave direction to bias, you would be fine. I like your method a lot, and if I could get it wrapped wrinkle free dry, I would do exactly what you have described. I’m with you on this.