  The $\alpha'$-expansion of string theory provides a rich set of
higher-dimension operators, indexed by $\zeta$ 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
$\zeta_2$-free operator insertions from the open superstring: $\zeta_3$,
$\zeta_5$, $\zeta_3^2$, and $\zeta_7$. We succeed in finding a representation
for the first three in terms of box, triangle, and bubble numerators. In the
case of $\zeta_7$ 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.
