It's very common for social scientists to have binary or ordinal indicators (e.g., yes/no interview items or likert-rated questionnaire items). It's also very common for them to model their ordered/non-ordered categorical data with structural equation model (SEM)/factor analytic (FA) techniques using weighted least estimators (e.g., WLS, robust WLS or WLSMV).
Yet, as far as I can tell, there is no established method for comparing two SEM/FA models which are not nested. With maximum likelihood estimators, people usually use AIC and BIC but they are not available for WLS estimation. I've seen some use fit statistics such as CFI, TLI, and RMSEA, but have not managed to find an empirically grounded criteria indicating a substantial change in model fit (e.g., compared to the less restricted model, an increase in RMSEA of say .05 suggests that the model fit worsened meaningfully).
Can anyone recommend an empirically credible criteria for model comparison using WLS estimation for non-nested models? If there is no empirically tested criteria, perhaps share your experience of what you see in the field.