I will address your second question: "one has to prove that the classification diagram functor is sent under this Quillen equivalence to something weakly equivalent to the coherent nerve".
The answer is yes, this is true. First note that the simplicial category $M^\circ$ is equivalent as a simplicial category to $L^HM$, the hammock localization of M (at the weak equivalences). See here for references. So the answer to your second question (or is it a remark?) follows as a special case of this previous MO answer (which cites work of Barwick-Kan and Toen). So the classification diagram functor and the coherent nerve of $M^\circ$ are sent to equivalent things under the Quillen equivalence between quasicategories and complete Segal spaces (I prefer the term Rezk categories or Rezk $\infty$-categories).
I believe that combined with your other comments (notably using Julie Bergener's results) this is enough to answer your main question in the affirmative.