The purpose of this paper is twofold. On one hand, we introduce a modification of the dual canonical basis for invariant tensors of the 3-dimensional irreducible representation of U_q(sl_2), given in terms of Jacobi diagrams, a central tool in quantum topology. On the other hand, we use this modified basis to study the so-called homotopy sl_2 weight system, which is its restriction to the space of Jacobi diagrams labeled by distinct integers. Noting that the sl_2 weight system is completely determined by its values on trees, we compute the image of the homotopy part on connected trees in all degrees; the kernel of this map is also discussed.