When data analysts use linear mixed models, they usually encounter two practical problems: (a) the true model is unknown and (b) the Gaussian assumptions of the errors do not hold. While these problems commonly appear together, researchers tend to treat them individually by (a) finding an optimal model based on the conditional Akaike information criterion (cAIC) and (b) applying transformations on the dependent variable. However, the optimal model depends on the transformation and vice versa. In this paper, we aim to solve both problems simultaneously. In particular, we propose an adjusted cAIC by using the Jacobian of the particular transformation such that various model candidates with differently transformed data can be compared. From a computational perspective, we propose a step-wise selection approach based on the introduced adjusted cAIC. Model-based simulations are used to compare the proposed selection approach to alternative approaches. Finally, the introduced approach is applied to Mexican data to estimate poverty and inequality indicators for 81 municipalities.