Is there a clear-cut analogue of the strong form of Serre's Conjecture for residually reducible Galois Representations?

Billerey and Menares have studied this question for the reducible representations $\bar{\rho} = 1 \oplus \chi_p^{k-1} $ in https://arxiv.org/abs/1309.3717 In this case the prime-to-$p$ part of the Artin conductor is 1, and it is not always the case that $\bar{\rho} $ arises from a cuspidal eigenform of weight $k $ and level $1$.

In addition to François' answer, here is what can be said if you only seek for an isomorphism between the semi-simplification of your residual representation $\overline{\rho}$ and the semi-simplification of the reduction of a $p$-adic representation attached a Hecke eigenform.

Consider an odd mod $p$ Galois representation $\overline{\rho}=\nu_1\oplus\nu_2$ of Serre weight $k$ and level $N$ (coprime to $p$) and assume $p>k+1$. Then, there exist $\epsilon_1,\epsilon_2$ two Galois characters unramified at $p$ such that $\overline{\rho}\simeq\epsilon_1\oplus\epsilon_2\chi_p^{k-1}$. Set $\eta=\epsilon_1^{-1}\epsilon_2$. Then, there is a newform $f$ of (optimal) weight $k$ and level $N$ and a prime ideal $\mathfrak{p}$ over $p$ in $\overline{\mathbf{Q}}$ such that we have $\overline{\rho}\simeq\overline{\rho}_{f,\mathfrak{p}}^{ss}$ if and only if we have $B_{k,\eta}=0$ or $\eta(\ell)\ell^k=1$ for some prime $\ell$ dividing $N$.

Here, $\eta(\ell)=\eta(\mathrm{Frob}_\ell)$ if $\eta$ is unramified at $\ell$ and $\eta(\ell)=0$ otherwise. (Roughly speaking $B_{k,\eta}$ is the mod $p$ reduction of the $k$-th Bernoulli number associated with a lift of $\eta$.)