In this paper, we introduce a family of Choquet-based non-associative aggregation functions for application in the fuzzy reasoning method proposed by Barrenechea et al. for fuzzy rule-based classification systems. The family is constructed manipulating the standard definition of Choquet Integral and substituting the product operator by the general definition of the family of overlap functions Cα(x; y) = xy(1 + α(1 - x)(1 - y)), for α ∈ [-1; 0[∪]0; 1], resulting in non-associative aggregation functions. A comparative study considering different values for α and the power measure (whose exponent is learned genetically to adapt it to each class) is presented. The approach is tested in seventeen numerical dataset selected from the KEEL dataset repository. We compare the obtained results with the work presented by Barrenechea et al., showing that the proposed approach can offer also good performance, so providing more flexibility to that proposal, enlarging its scope of applications.