Counterexample in two dimensions, using complex notation: $\omega=\{z\in\mathbb C:|z|<1\}$, $\Omega=\{z\in\mathbb C:|z|<3\}$, $\theta(z)=\mathrm{Re}\,\frac{1}{z-2}$.
Indeed, suppose $\tilde \theta$ exists. Since it is in $W^{2,\infty}(\Omega)$, its second-order weak derivatives are functions that can be calculated by differentiating $\tilde\theta$ pointwise. In particular, $\Delta \tilde\theta=0$ holds in the weak sense because it holds a.e. By Weyl's lemma $\tilde \theta $ is harmonic in $\Omega$ in the classical sense. But then it must be different from $\mathrm{Re}\,\frac{1}{z-2}$ somewhere in $\Omega$, contradicting the uniqueness theorem for harmonic functions. QED
The obstruction lies in that $W^{2,\infty}$ requirement prescribes both the values and the normal derivative of $\tilde \theta$ on $\partial\omega$. This means you are trying to solve the Cauchy problem for an elliptic equation, famously shown to be ill-posed by Hadamard.