As far as I understand just-identified model has zero degrees of freedom and its model fit indices do not make much sense. On the contrary, overidentified models have meaningful chi-square, CFI, etc. Some researchers compare just-identified models to overidentified ones using these fit indices. 

I wonder whether it is justifiable to compare fit indices (such as CFI, RMSEA, SRMR, chi-square) between just-identified and overidentified models given these models are nested?

I know I can do it using AIC and BIC. But what about CFI and other ones?