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The $α'$-expansion of string theory provides a rich set of higher-dimension operators, indexed by $ζ$ values, which can be used to study color-kinematics duality and the double copy. These two powerful properties, actually first noticed in tree-level string amplitudes, simplify the construction of both gauge and gravity amplitudes. However, their applicability and limitations are not fully understood. We attempt to construct a set of color-kinematics dual numerators at one loop and four points for insertions of operator combinations corresponding to the lowest four $ζ_2$-free operator insertions from the open superstring: $ζ_3$, $ζ_5$, $ζ_3^2$, and $ζ_7$. We succeed in finding a representation for the first three in terms of box, triangle, and bubble numerators. In the case of $ζ_7$ we find an obstruction to a fully color-dual representation related to the bubble-on-external-leg type diagrams. We discuss two paths around this obstruction, both of which signal an incompatability between color-kinematics duality and manifesting certain desired properties. Using the constructed color-dual numerators, we find two different Bern-Carrasco-Johansson double copies that produce candidate closed-string-insertion numerators. Both approaches to the double copy match the kinematics of the cuts, with relative normalization set by either summing over both double copies including degeneracy or by including an explicit prefactor on the double-copy numerator definitions.