Hi,

It's incredibly important to understand what your data look like (checking for skew and kurtosis, both uni- and multivariate), but at the end of the day I think it's easiest to just use test statistics that account for non-normality. A lot of programs will provide these just by clicking the appropriate options.

I do SEM in EQS and often use Robust ML (Maximum likelihood; see Satorra-Bentler, 1990) with N~400-500 and Mardia Coeff's anywhere from 3 to 200. I'm a grad student at UCLA and Peter Bentler himself (a faculty member at UCLA) has looked over manuscripts of mine in which I report these stats/approach and it gets his approval. So, for whatever that's worth. For a more formal source to cite, Ullman and Bentler (2013) report data analyzed (albeit as a descriptive example) with ML Robust when Mardia's = 238.65. Citations below.

You could play around with transformations, but as the other post mentions, they may not fix the problem anyways. And in terms of publications, unfortunately I think there's still some stigma around transformations (as though you've done something sneaky to make your data look better, which is completely unfounded). So, again, I think it's easiest just to mention that there was considerable variance among each of your variables but the data were not multivariate normally distributed, so you used robust maximum likelihood estimation (cite Satorra-Bentler, 1990)

Satorra, A., & Bentler, P. M. (1990). Model conditions for asymptotic robustness in the analysis of linear relations. Computational Statistics & Data Analysis, 10, 235-249.

Ullman, J. B., & Bentler, P. M. (2013). Structural equation modeling. In J. A. Schinka & W. F. Velicer (Eds.), Handbook of Psychology, Vol 2: Research Methods in psychology (pp. 661-690). Hoboken NJ: Wiley.