I am conducting confirmatory LPA to compare 4 non-nested models. I use BIC and entropy to select the best-fitting model. However, these statistics do not converge. The lowest-BIC model has lower entropy (.74) than the second-lowest-BIC model (.79). As such, I am not clear which model to select as the best-fitting model.
I wonder: 1. Should I attach more weight to either BIC or entropy? 2. Would it be helpful to look at additional indices, such as aBIC or AIC?
You should go with BIC. Only if 2 models have very similar BIC would I go with entropy. I think of entropy as R-square in SEM while BIC is closer to a model fit index - R-square has nothing to do with model fit but only the model's usefulness (which you wouldn't benefit from if the model doesn't fit well).
You can always work with what I call neighboring models - that is, for a given H0 model, you can specify a more general model (e.g. residual covariances) and see if those extra parameters are significant.