A full description is provided for the BV -structure on the Hochschild cohomology of exceptional local algebras of quaternion type determined by parameters (k, 0, d) in the case of an even parameter k ≥ 3, according to Erdmann’s classification. The method of comparison morphisms and the weak self-homotopy method are developed and used. This article may be viewed as a generalization of similar results about the BV -structures on the Hochschild cohomology of algebras of quaternion type. © 2024 American Mathematical Society