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This should work according to the embedding $$W \hookrightarrow L^{\frac2{1-2s}}((L^2(\Omega),H^2(\Omega))_{\theta,1})$$ where $0 < s < \frac12$ and $0 \leq \theta < 1-s$ as in Amann: Linear parabolic problems involving measures, Theorem 3. (My calculations gave that your desired embedding is correct for all $r \in [2,2+\frac4n)$ which corresponds to $s < \frac1{2+n}$ and $\theta = \frac12(ns+1)$, but of course you should double-check that..)