= Solution
Use the physical <refractive index> convention $k=k_0n$, with $k_0$ the background <wavenumber> and time dependence $e^{-i\omega t}$. Treat $B$ as a bounded region in $\mathbb R^3$; the printed $B\in\mathbb R^3$ should be $B\subset\mathbb R^3$. Define the contrast $q=n^2-1=2n_\epsilon+n_\epsilon^2$ and the outgoing <Green function>
$$
G_{k_0}(\mathbf r,\mathbf r')=\frac{e^{ik_0|\mathbf r-\mathbf r'|}}{4\pi|\mathbf r-\mathbf r'|},\qquad
(\Delta+k_0^2)G_{k_0}=-\delta.
$$
The <Sommerfeld radiation condition> selects this sign of the outgoing <wave phase>. Since $(\Delta+k_0^2)\psi=-k_0^2q\psi$, the <Lippmann-Schwinger equation> is
$$
\psi=\psi_i+T_q\psi,\qquad
(T_q\phi)(\mathbf r)=k_0^2\int_BG_{k_0}(\mathbf r,\mathbf r')q(\mathbf r')\phi(\mathbf r')\,d\mathbf r'.
$$
The positive sign in this integral follows from the minus sign in the defining <Green function> equation. One may instead define the <scattering potential> with the opposite sign, provided both equations change consistently.
If $\|T_q\|<1$ on the chosen field space inside $B$, its <Neumann series> converges. The <Born series> for the scattered field is
$$
\boxed{\psi_s=\psi-\psi_i=\sum_{m=1}^\infty T_q^m\psi_i.}
$$
For observation points outside $B$, restrict the intermediate factors to $B$ and use the same outgoing integral for the final factor. For example, the first two terms are
$$
\begin{aligned}
\psi_s^{(1)}(\mathbf r)&=k_0^2\int_BG_{k_0}(\mathbf r,\mathbf r_1)q(\mathbf r_1)\psi_i(\mathbf r_1)\,d\mathbf r_1,\\
\psi_s^{(2)}(\mathbf r)&=k_0^4\int_B\int_BG_{k_0}(\mathbf r,\mathbf r_1)q(\mathbf r_1)G_{k_0}(\mathbf r_1,\mathbf r_2)q(\mathbf r_2)\psi_i(\mathbf r_2)\,d\mathbf r_2\,d\mathbf r_1.
\end{aligned}
$$
The $m$th term describes $m$ successive scattering interactions. The first <Born approximation for scalar wave scattering> keeps one interaction and replaces the field inside the medium by the incident field. Higher terms describe <multiple scattering> and the resulting feedback on the internal field.
A concrete sufficient condition follows on $L^\infty(B)$. If $d_B$ is the diameter of $B$, then for $\mathbf r\in B$ the region lies in the ball of radius $d_B$ about $\mathbf r$, and
$$
\|T_q\|_{\infty\to\infty}\leq k_0^2\|q\|_\infty\sup_{\mathbf r\in B}\int_B\frac{d\mathbf r'}{4\pi|\mathbf r-\mathbf r'|}
\leq\frac12k_0^2d_B^2\|q\|_\infty=:\rho.
$$
Thus $\boxed{\rho\ll1}$ is a conservative sufficient condition for the first <Born approximation for scalar wave scattering>. When $\rho<1$, the omitted internal-field terms satisfy
$$
\|\psi_s-T_q\psi_i\|_\infty\leq\frac{\rho^2}{1-\rho}\|\psi_i\|_\infty.
$$
The bound ignores cancellation in the oscillatory <Green function>, so it is sufficient, not necessary. A common physical small-contrast criterion for an extended weak medium is small accumulated extra <wave phase>, $k_0\ell\|n_\epsilon\|_\infty\ll1$, together with weak scattering and no resonant internal enhancement. Small local contrast alone is not enough for an arbitrarily large or resonant object.
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