Hi xralphyx,

Certainly, hypothesis testing can be influenced by sample size, though your p values seem to be considerably low. You can either try other multivariate normality test or try by checking the assumption graphically. I would recommend the latter since this way you can also get some idea about what the problem may be. In order to detect multivariate normality with a graph, plot the squared Mahalanobis Distance against the quantiles of a Chi-square distribution with the degrees of freedom equal to the number of variables. There's some info about that graph in this link. With that, you can understand more deeply the distribution of your data.

Now, you are incredibly right when you say that multivariate normality it is a important assumption in SEM and should be addressed. When there are violations, there are other models available, like PLS models, which don't require multivariate normality, as far as I recall. Also, there are corrections that can be applied to the estimators in order to improve the results. And of course, you can transform your data (yet that may affect interpretations). The way to go will depend on what you are measuring and the type of distribution you have.

Hope this helps!