Under what (boundary) conditions is it possible to conclude a weak solution to $\Delta u = f$ which is also in $H^2_0$ of this problem?
The answer is that: no boundary conditions can do this.
Rule of thumb: If the imposed boundary condition leads to a well-posed weak formulation (solution exists and is unique), then no matter what this boundary condition is, we can not get a weak solution $u\in H^2_0$ in general.
First notice $H^2_0$ means:
$$
H^2_0(U) := \{u\in H^2: \; u\big|_{\partial U} = 0, \; \nabla u\big|_{\partial U} = 0\}.
$$
Also note if $u$ is a constant on boundary, then $u$ does not change tangentially along any curve on boundary, so the directional derivative of $u$ along any tangent vector of $\partial U$ is 0. The only thing we wanna deal with is the normal derivative:
$$
u\big|_{\partial U} = 0 \text{ and } \nabla u\big|_{\partial U} = 0 \implies u\big|_{\partial U} = 0\text{ and } \frac{\partial u}{\partial n}\Big|_{\partial U} = 0
$$
Now if $u$ is a weak solution of the Poisson equation, and $u\in H^2_0$, this means
$$\left\{
\begin{aligned}
-\Delta u &= f \quad \text{ in } U,
\\
u &= 0 \quad \text{ on } \partial U,
\\
\partial u/\partial n &= 0 \quad \text{ on } \partial U,
\end{aligned}\right.\tag{$\star$}
$$
for some $f$.
However, $(\star)$ is not a well-posed problem. Looking for a function $u\in H^2_0$ is like imposing Dirichlet and Neumann boundary conditions at the same time (a Cauchy boundary problem), which can be proved ill-posed in several ways.
A simple argument can be: if neglecting Neumann boundary at first, Poisson equation together with the Dirichlet boundary has a unique solution for any $f\in L^2(U)$. This means $\nabla u$ can be evaluated a.e. in $U$, and if $\partial u/\partial n|_{\partial U}$ is defined as a limit from the interior to the boundary in the direction of the normal , then $\partial u/\partial n$ is already determined in this case.
In general we can't expect $\partial u/\partial n = 0$ coincides with what you obtained from the solution of Dirichlet boundary problem.
A counterexample would be: suppose we have a classical solution which coincides with a strong and weak solution to the Dirichlet boundary problem except a measure zero set, and $f\leq 0$. By maximum principle the maximum is achieved on boundary, $u|_{\partial U}=0> u(x)$ for $x\in U$, then $\partial u/\partial n$ must be strictly greater than 0 pointwisely (see Gilbarg and Trudinger Page 32).
Another argument would be if $\partial u/\partial n|_{\partial U} = 0$, by divergence theorem:
$$
\int_U f\,dx = \int_U \mathrm{div}(\nabla u)\,dx = \int_{\partial U} \frac{\partial u}{\partial n} \,dS = 0,
$$
which means $f$ has average 0 in $U$. Hence this says that, if $f$ does not have zero average, the $(\star)$ does not have a solution.
Some comment about your terminology, "regularization" doesn't mean what you have written in your question, what you applied is just elliptic regularity.
