This problem was already mentioned in the original 't Hooft-Veltman article and solved by Breitenlohner and Maison. This solution is known by the name "HVBM scheme" (after 't Hooft, Veltman, Breitenlohner and Maison).
A clear description of this regularization procedure is given for example in the following dissertation by Barbara Jäger. It consists basically of splitting the metric into a $4$-dimnsional and $(d-4)-$ parts, assuming the Levi-Civita tensor to have non-vanishing components only in the 4-dimensional subspace. In addition $\gamma_5$ is assumed to anti-commute with the $\gamma$ matrices of the 4-dimnsional subspace and commutes with the others. This procedure leads to consistent Ward identities.
As mentioned in Jäger 's thesis, this procedure leads to a higher complexity in the computation of the Feynman diagrams, but there exist computer algebra programs implementing this scheme.