  We construct new representations of tree-level amplitudes in D-dimensional
gauge theories with deformations via higher-mass-dimension operators $\alpha'
F^3$ and $\alpha'^{2} F^4$. Based on Berends-Giele recursions, the tensor
structure of these amplitudes is compactly organized via off-shell currents. On
the one hand, we present manifestly cyclic representations, where the
complexity of the currents is systematically reduced. On the other hand, the
duality between color and kinematics due to Bern, Carrasco and Johansson is
manifested by means of non-linear gauge transformations of the currents. We
exploit the resulting notion of Bern-Carrasco-Johansson gauge to provide
explicit and manifestly local double-copy representations for gravitational
amplitudes involving $\alpha' R^2$ and $\alpha'^2 R^3$ operators.
