Green's functions can be calculated once, then used repeatedly for the same configuration, but with different input functions (e.g., $\rho(\mathbf{x})$). Considering that the fundamental Green's function (cf, the table in [1]), $G_F$, is
$$ G_F(\mathbf{x};\mathbf{x}^\prime) =
\frac{-1}{4\,\pi\| \mathbf{x} \|} \,, $$ I consider
\begin{equation}
\tag{1}
G(\mathbf{x};\mathbf{x}^\prime) =
\frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|} +
\frac{k_{1 }}{\|k_{2 }\mathbf{x} - k_{3 }\mathbf{x}^\prime\|} \,,
\end{equation}
where the consants obey the restriction that $
k_{2 } \neq k_{3 }\,.$
Now, I demonstrate that there exists some $k_{1 } $, $k_{2 }$, and $k_{3 }$ such that the function in (1) satisfies the four requirements for the calculation of a Green' function [2].
First Requirement
The first requirement is that $L[G(\mathbf{x};\mathbf{x}^\prime)]=0$ except at
$$\mathbf{x}= \mathbf{x}^\prime\quad\text{and}\quad \mathbf{x}=\frac{k_{3}}{k_{2}}\,\mathbf{x}^\prime.$$
It is tedious to take the Laplacian of the fundamental Green's functions. It is no more tedius to take the Laplacian of each term of the Green's function in (1). One can take the Laplacian by hand or with a symbolic software package. Needless to say, both the fundamental Green's function and Green's function here satisfy the first requirement.
Second Requirement
The second requirement is that the Green's function satisfy linear homogeneous boundary conditions
$$B_j[G(\mathbf{x};\mathbf{x}^\prime)] =0,\quad\text{for}~j=1,2,\ldots ,m\,.$$
For the purpose of calculating a Green's function, the requirements must pertain to linear homogeneous boundary conditions. These can be either homogeneous Dirichlet boundary conditions, homogeneous Neumann boundary conditions, or homogeneous mixed boundary conditions. Distinct homogeneous boundary conditions (i.e., Dirchilet, Neumann, or mixed) result in the calculation of distinct Green's functions.
In the OP's problem here, there are two boundary conditions (i.e., $B_1[G(\mathbf{x};\mathbf{x}^\prime)]$ and $B_2[G(\mathbf{x};\mathbf{x}^\prime)]$). The first boundary condition implicit in the boundary-value problem here (and implicit in many others problems) is a homogeneous Dirchilet boundary condition in the limit as $\left\|\mathbf{x} \right\| $ goes to infinity. Thus,
$$B_1[G(\mathbf{x};\mathbf{x}^\prime)] = \lim_{\left\|\mathbf{x}\right\|\to\infty}G(\mathbf{x};\mathbf{x}^\prime) = 0.$$
By inspection, the fundamental Green's function as well as the Green's function in (1) meet this requirement.
The second boundary condition more or less explicit in the boundary-value problem here is a homogeneous Dirchilet boundary condition for $\left\|\mathbf{x}\right\| =a$. Thus,
$$B_2[G(\mathbf{x};\mathbf{x}^\prime)] = \left.G(\mathbf{x};\mathbf{x}^\prime)\right\|_{\left\|\mathbf{x}\right\| =a} = 0.$$
By inspection, the fundamental Green's function does not meet this requirement. However, the Green's function in (1) would meet this requirement if
\begin{align}
\tag{2}
k_1 &= \frac{1}{4\,\pi},\qquad
k_2 = \frac{\left\|\mathbf{x}^\prime\right\|}{a}
,
\qquad
k_3 = \frac{a}{\left\|\mathbf{x}^\prime\right\|}\,.
\end{align}
In such case (1) becomes,
\begin{equation}
G(\mathbf{x};\mathbf{x}^\prime) =
\frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|} +
\frac{1}{4\,\pi\|\frac{\left\|\mathbf{x}^\prime\right\|}{a}\mathbf{x} - \frac{a}{\left\|\mathbf{x}^\prime\right\|}\mathbf{x}^\prime\|} \,,
\end{equation}
Third Requirement
The third requirement is that since the Laplacian is a second-order differential operator, the $G$ must be continuous at $\mathbf{x} = \mathbf{x^\prime}$.
Note that:
\begin{align*}
&
\lim_{ \boldsymbol{\epsilon} \to\mathbf{0}}
\left[\frac{-1}{4\,\pi\left\| \mathbf{x} - \mathbf{x}^\prime \right\| }\right]_{\mathbf{x} = \mathbf{x^\prime}- \boldsymbol{\epsilon}}^{\mathbf{x} = \mathbf{x^\prime} + \boldsymbol{\epsilon}}
+
\left[\frac{k_{1i}}{\left\| k_{2i}\,\mathbf{x} - k_{3i}\,\mathbf{x}^\prime \right\| }\right]_{\mathbf{x} = \mathbf{x^\prime}- \boldsymbol{\epsilon}}^{\mathbf{x} = \mathbf{x^\prime} + \boldsymbol{\epsilon}}
\\
&\quad=
\lim_{ \boldsymbol{\epsilon} \to\mathbf{0}}
\left[
\frac{-1}{4\,\pi\left\| \left[ \mathbf{x^\prime}+ \boldsymbol{\epsilon}\right] - \mathbf{x}^\prime \right\| }
+
\frac{k_{1}}{4\,\pi\left\| \left[ \mathbf{x^\prime}- \boldsymbol{\epsilon}\right] - \,\mathbf{x}^\prime \right\| }
\right]
\\
&\quad+
\lim_{ \boldsymbol{\epsilon} \to\mathbf{0}}
\left[
\frac{k_{1i}}{\left\| k_{2i}\,\left[ \mathbf{x^\prime}+ \boldsymbol{\epsilon}\right] - k_{3i}\,\mathbf{x}^\prime \right\| }
-
\frac{k_{1i}}{\left\| k_{2i}\,\left[ \mathbf{x^\prime}- \boldsymbol{\epsilon}\right] - k_{3i}\,\mathbf{x}^\prime \right\| }
\right]
\\
&\quad=
\lim_{ \boldsymbol{\epsilon} \to\mathbf{0}}
\left[
\frac{-1}{4\,\pi\left\| \boldsymbol{\epsilon} \right\| }
+
\frac{1}{
4\,\pi\left\|
- \boldsymbol{\epsilon}
\right\| }
\right]
\\
&\quad+
\lim_{ \boldsymbol{\epsilon} \to\mathbf{0}}
\left[
\frac{k_{1i}}{\left\| k_{2i}\, \boldsymbol{\epsilon} + \mathbf{x^\prime} \left[k_{2i} - k_{3i}\right] \right\| }
-
\frac{k_{1i}}{
\left\|
-k_{2i}\, \boldsymbol{\epsilon}
+
\mathbf{x^\prime}\left[ k_{2i} -k_{3i} \right]
\right\| }
\right]
\\
&\quad=
\frac{-1}{\left\| \mathbf{x^\prime} \left[k_{2i} - k_{3i}\right] \right\| }
-
\frac{k_{1i}}{
\left\|
\mathbf{x^\prime}\left[ k_{2i} -k_{3i} \right]
\right\| }
\\
&\quad=
\frac{k_{1i}}{\left| k_{2i} - k_{3i}\right|\left\| \mathbf{x^\prime} \right\| }
-
\frac{k_{1i}}{
\left| k_{2i} - k_{3i}\right|
\left\|
\mathbf{x^\prime}
\right\| }
\\
&\quad =0\,.
\end{align*}
Since the evaluation of this limit is identically zero, therefore $G$ is continuous at $\mathbf{x} = \mathbf{x^\prime}$.
Fourth Requirement
The fourth requirement is that
$\lim_{\boldsymbol{\epsilon}\to \mathbf{0}}\int_{\mathbf{x}^\prime - \boldsymbol{\epsilon}}^{\mathbf{x}^\prime + \boldsymbol{\epsilon}} L\left[G(\mathbf{x}-\mathbf{x}^\prime)\right]\,dV = 1$.
To evaluate this, I first must denote two geometrical objects. These are a ball and the surface of a ball. By $\mathcal{B}{\left(\mathbf{0},\epsilon\right)}$ I denote a ball with center at $\mathbf{0}$ and radius $\epsilon$. By $\mathcal{S}{\left(\mathbf{0},\epsilon\right)} $ I denote the closed spherical surface with center at $\mathbf{0}$ and radius $\epsilon$ (i..e, the boundary of the ball $\mathcal{B}{\left(\mathbf{0},\epsilon\right)}$).
With these geometrical objects the fourth requirement is written as
$$\lim_{\varepsilon\to0}\iiint_{\mathcal{B}{\left(\mathbf{x}^\prime,\epsilon\right)}} L[G{\left(\mathbf{x};\mathbf{x}^\prime\right)}] \,\,d^3\mathbf{x}
= 1\,.$$
Note that (1) is composed of two terms:
\begin{equation}
\frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|} \qquad \text{and}\qquad
\frac{k_{1 }}{\|k_{2 }\mathbf{x} - k_{3 }\mathbf{x}^\prime\|} \,.
\end{equation}
Since the evaluation of the volume integral is done only in the neighborhood around $\mathbf{x}^\prime$ and since the Laplacian of the second terms is zero in that neighborhood (see the first requirement) we have that
\begin{align*}
\lim_{\varepsilon\to0}\iiint_{\mathcal{B}{\left(\mathbf{x}^\prime,\epsilon\right)}} L[G{\left(\mathbf{x};\mathbf{x}^\prime\right)}] \,\,d^3\mathbf{x}
=
\lim_{\varepsilon\to0}\iiint_{\mathcal{B}{\left(\mathbf{x}^\prime,\epsilon\right)}} \boldsymbol{\nabla}\cdot\left[\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|}\right] \,d^3\mathbf{x}
\end{align*}
Now, I do a change of variables. I write that
$\boldsymbol{\chi} = \mathbf{x} - \mathbf{x}^\prime$. Thus,
\begin{align*}
\lim_{\varepsilon\to 0}\iiint_{\mathcal{B}{\left(\mathbf{x}^\prime,\epsilon\right)}} \boldsymbol{\nabla}\cdot\left[\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|}\right] \,d^3\mathbf{x}
&=
\lim_{\varepsilon\to0}\iiint_{\mathcal{B}{\left(\mathbf{0} ,\epsilon\right)}} \boldsymbol{\nabla}\cdot\left[\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \boldsymbol{\chi} \|}\right] \,d^3\boldsymbol{\chi}
\end{align*}
Now, I use the divergence theorem to write that
\begin{align*}
\lim_{\varepsilon\to 0}\iiint_{\mathcal{B}{\left(\mathbf{x}^\prime,\epsilon\right)}} \boldsymbol{\nabla}\cdot\left[\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|}\right] \,d^3\mathbf{x}
&=
\lim_{\varepsilon\to0}\iint_{\mathcal{S}{\left(\mathbf{0} ,\epsilon\right)}} \left[\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \boldsymbol{\chi}{\left(\rho,\theta,\varphi\right)} \|}\right] \cdot d\mathbf{S}
\end{align*}
I know that $\| \boldsymbol{\chi}{\left(\rho,\theta,\varphi\right)} \| = \rho$. Thus,
$\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \boldsymbol{\chi}{\left(\rho,\theta,\varphi\right)} \|} = \boldsymbol{\nabla} \frac{1}{4\,\pi \,\rho^2}$. Further, since I am integrating over a sphere of radius $\epsilon$,
$$
\left(\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \boldsymbol{\chi}{\left(\rho,\theta,\varphi\right)} \|}\right)_{\mathcal{S}{\left(\mathbf{0} ,\epsilon\right)}} = \frac{1}{4\,\pi \,\epsilon^2},$$
and
the surface element, $d\mathbf{S} $, evaluated on the surface of the ball is given by the equation
$$
\left(d\mathbf{S}\right)_{\mathcal{S}{\left(\mathbf{0} ,\epsilon\right)}} =\epsilon^2\,\sin\theta \,d\theta \,d\varphi \,\hat{\boldsymbol{\rho}}.$$
Consequently,
\begin{align*}
\lim_{\varepsilon\to 0}\iiint_{\mathcal{B}{\left(\mathbf{x}^\prime,\epsilon\right)}} \boldsymbol{\nabla}\cdot\left[\boldsymbol{\nabla} \frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|}\right] \,d^3\mathbf{x}
&=
\frac{1}{4\,\pi} \iint_{\mathcal{S}{\left(\mathbf{0} ,\epsilon\right)}}
\sin\theta \,d\theta \,d\varphi
\end{align*}
Finally, since
$ \iint_{\mathcal{S}{\left(\mathbf{0} ,\epsilon\right)}}
\sin\theta \,d\theta \,d\varphi= 4\,\pi
$, we have shown that
the function in (1) satisfies the fourth requirement.
Subject to the constraints in (2), the function in (1) satisfies the four requirements necessary to construct a Green's function. Namely, the Green's function for this situation is given by the equations
\begin{equation}
\boxed{
G(\mathbf{x};\mathbf{x}^\prime) =
\frac{-1}{4\,\pi\| \mathbf{x} - \mathbf{x}^\prime\|}
+
\frac{a\left\|\mathbf{x}^\prime\right\|}{4\,\pi\left\| \left\|\mathbf{x}^\prime\right\|^2 \mathbf{x}
- a^2\,\mathbf{x}^\prime\right\|} \,.} \end{equation}
This Green's function satisfies the specific homogeneous Dirichlet boundary conditions that are necessary for this situation. Thus, this Green's function is appropriate for this situation irrespective of the form of the input (e.g., $\rho{(\mathbf{x})}$).
Bibliography
[1] https://en.wikipedia.org/wiki/Green%27s_function
[2] Zwillinger, Handbook of Differential Equations, Fourth Edition, p. 210.
