I am not sure how you define the pseudo-inverse of an operator between infinite dimensional spaces, but but here is what we know:

The operator $A$ is a continuous operator from $H^1(\Omega)$ to the mean value free subspace of its dual. This subspace is closed, such that the closed range theorem applies: there is a continuous operator $B$ from this subspace to $H^1(\Omega)$, such that $AB f = f$ for all $f$ in this subspace.

Note that $B$ is not uniquely defined. In order to select $A^\dagger$ from the possible choices for $B$, you can for instance require that for any $f$ holds \begin{gather} \int_\Omega Af\,dx = 0. \end{gather}

The operator $A$ is a continuous operator from $H^1(\Omega)$ to the mean value free subspace of its dual. This subspace is closed, such that the closed range theorem applies: there is a continuous operator $B$ from this subspace to $H^1(\Omega)$, such that $AB f = f$ for all $f$ in this subspace.

Note that $B$ is not uniquely defined. In order to select $A^\dagger$ from the possible choices for $B$, you can for instance require that for any $f$ holds \begin{gather} \int_\Omega Af\,dx = 0. \end{gather}