For $u \in H^2(I)\cap H^1_0 (I)$, $I=(1,2)$, the inequality is equivalent to $$\|(-\Delta)^{-s/2}v\|_\infty \le C \|v\|_2, \quad v \in L^2(I).$$ Let us use
$$
(-\Delta)^{-s/2}v=\frac{1}{\Gamma (s/2)}\int_0^\infty t^{s/2 -1}T(t)v\, dt$$
where $T(t)$ is the semigroup generated by the Dirichlet Laplacian in $I$. If $\lambda_1$ is its first eigenvalue then $\|T(t)\|_{2 \to 2} \le e^{-\lambda_1 t}$ and $\|T(t)\|_{ 2 \to \infty} \le C_1t^{-1/4}$, by Gaussian estimates. The semigroup law yields $\|T(t)\|_{2 \to \infty} \le C_2 t^{-1/4}e^{-\lambda_1 t/2}$ and finally
$$\|(-\Delta)^{-s/2} v\|_\infty \le \frac{C_2}{\Gamma (s/2)}\|v\|_2 \int_0^\infty t^{s/2-5/4}e^{-\lambda_1 t/2}\, dt \le C_3 \|v\|_2$$
if $1/2<s<2$. Finally, note that all constants $C_i$ can be computed since $\lambda_1$ is known and the Gaussian estimates follow from domination with the heat semigroup in the whole space.