using Hausman test for endogeneity to assess SEM model

I've searched the internet and textbooks, but cannot find an answer to this question. Please help me out! It's a final step in my thesis.

I have a SEM path model, consisting of all direct effects, and one mediator path. This mediator path is included in my SEM to control for the simultaneous estimation of X and Y.

My thesis supervisor told me to run an additional endogeneity test, preferably a Hausman test. This Hausman works perfectly fine, indicating no endogeneity problem. But Hausman tests whether the OLS (!) coefficients are unbiased, while my theory is a SEM-like path model. Can I use this test?

By the way, if I run an OLS regression model of my theory (i.e., without the mediating-path), I get both qualitatively and quantitatively similar results.