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..)

This should work according to the embedding $$W \hookrightarrow L^{\frac2{1-2s}}\bigl((L^2(\Omega),H^2(\Omega))_{\theta,1}\bigr)$$ where $0 < s < \frac12$ and $0 \leq \theta < 1-s$ as in Amann: Linear parabolic problems involving measures, Theorem 3.

To elaborate a bit more: By a relation between real and complex interpolation spaces, interpolation identitites for Bessel potential spaces and the supposed smoothness of $\Omega$, $$\bigl(L^2(\Omega),H^2(\Omega)\bigr)_{\theta,1} \hookrightarrow \bigl[L^2(\Omega),H^2(\Omega)\bigr]_\theta = H^{2\theta}(\Omega),$$ and $H^{2\theta}(\Omega) \hookrightarrow W^{1,r}(\Omega)$ where $2\theta - \frac{n}2 \geq 1-\frac{n}{r}$. (The standard textbooks by Triebel, there Thms. 1.10.3.1 and 4.3.1.2 and 4.6.2, and Bergh/Löfström contain this.) Now it remains to connect $r$ and the integrability $2/(1-2s)$ from the above embedding.

(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..)