  We study CPT-odd non-minimal Lorentz-symmetry violating couplings in the
electroweak sector modifying the interactions between leptons, gauge mediators
and the Higgs boson. The tree-level (differential) cross sections for three
important electroweak processes are discussed: $e^+ \, e^- \rightarrow Z \, H$,
$e^+ \, e^- \rightarrow Z \, Z$ and $\gamma \, \gamma \rightarrow W^+ \, W^-$.
By considering next-generation $e^+ \, e^-$ colliders reaching center-of-mass
energies at the TeV scale and the estimated improved precision for the
measurements of the respective cross sections, we are able to project upper
bounds on the purely time-like background 4-vector as strict as $\lesssim
10^{-5} \, \mbox{GeV}^{-1}$, in agreement with previous work on similar
Lorentz-violating couplings.
