If $\mathrm{Gr}(B\Omega C)\xrightarrow{\backsim} B\Omega(\mathrm{Gr} C)$ actually means they both are not isomorphic but there is a canonical quasi-isomorphism from $\mathrm{Gr}(B\Omega C)$ to $B\Omega(\mathrm{Gr} C)$, I may conclude a similar statement from an argument of spectral sequences, see A primer on spectral sequences, example 4.3 (a homomorphism $B\Omega(\mathrm{Gr}C)\rightarrow \mathrm{Gr}(B\Omega C)$ that induces an isomorphism in homology).

If $\mathrm{Gr}(B\Omega C)\xrightarrow{\backsim} B\Omega(\mathrm{Gr} C)$ actually means they both are not isomorphic but there is a canonical quasi-isomorphism from $\mathrm{Gr}(B\Omega C)$ to $B\Omega(\mathrm{Gr} C)$, I may conclude a similar statement from an argument of spectral sequences, see A primer on spectral sequences, example 4.3 (a homomorphism $B\Omega(\mathrm{Gr}C)\rightarrow \mathrm{Gr}(B\Omega C)$ that induces an isomorphism in homology). I have no idea about an homomorphism in the opposite direction.