It is interesting to view your question above not only in light of the Goldstone-Wilczek (G-W) approach (G-W has provided a method for computing the fermion charge induced by a classical profile), but also by computing $1/2$-fermion charge found by [Jackiw-Rebbi](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.13.3398) using G-W method. For simplicity, let us consider the 1+1D case, and let us consider the $Z_2$ domain wall and the $1/2$-charge found by Jackiw-Rebbi. The construction, valid for 1+1D systems, works as follows.

Consider a Lagrangian describing spinless fermions $\psi(x)$ coupled to a classical background profile $\lambda(x)$

via a term $\lambda\,\psi^{\dagger}\sigma_{3} \psi$. In the high temperature phase, the v.e.v. of $\lambda$ is zero and no mass is generated

for the fermions. In the low temperature phase, the $\lambda$ acquires two degenerate vacuum values $\pm \langle \lambda \rangle$

that are related by a ${Z}_2$ symmetry. Generically we have

$$

\langle \lambda \rangle\,\cos\big( \phi(x) - \theta(x) \big),

$$

where we use the bosonization dictionary

$

\psi^{\dagger}\sigma_{3} \psi

\rightarrow

\cos(\phi(x))

$

and a phase change $\Delta\theta = \pi$ captures the existence of a domain wall

separating regions with opposite values of the v.e.v. of $\lambda$.

From the fact that the fermion density

$$

\rho(x)

=

\psi^{\dagger}(x)\psi(x)

=

\frac{1}{2\pi} \partial_{x} \phi(x)

,$$

and the current

$$J^\mu=\psi^{\dagger}\gamma^\mu\psi =\frac{1}{2\pi}\epsilon^{\mu \nu} \partial_\nu \phi,$$

it follows that the induced charge $Q_{\text{dw}}$ on the kink by a domain wall is

$$

Q_{\text{dw}}

=

\int^{x_0 + \varepsilon}_{x_0 - \varepsilon}\,dx\,\rho(x)

=

\int^{x_0 + \varepsilon}_{x_0 - \varepsilon}\,dx\,\frac{1}{2\pi} \partial_{x} \phi(x)

=

\frac{1}{2\pi} \pi =

\frac{1}{2},

$$

where $x_0$ denotes the center of the domain wall.

You can try to extend to other dimensions, but then you may need to be careful and you may not be able to use the bosonization.

See more details of the derivation here in the [page 13](https://arxiv.org/abs/1403.5256) of [this paper](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.195134).