Latent Growth Curve and covariates

#1
I'm examining latent growth curves in Mplus. My dependent variable is continuous, measured over three time points. I have a great unconditional model, and now I want to examine multiple time-invariant predictors on the latent intercept and slope. If I add the predictors, how do I evaluate if the predictors significantly improved model fit? Is this a nested model in which the chi-square difference test is appropriate, or is using BIC and AIC more appropriate?

Also, if using AIC and BIC, theoretically, if the predictors are important to the model, the BIC and AIC values of the model with the predictors will be lower than the unconstrained model?

Thanks for any help :)