If $N\geq 1$ is an integer not divisible by $p$, one can see that any system of Hecke eigenvalues $(a_\ell)$ arising from $S_k(\Gamma_1(N))$ is congruent mod $p$ to a system $(b_\ell)$ arising from $S_2(\Gamma_1(Np^n))$, for some $n$, using an interplay between a theorem of Serre (describing a purely mod $p$ Jacquet-Langlands correspondence), and the more classical, characteristic zero J-L between ${\rm GL}_2$ and the multiplicative group $G$ of the $\mathbf{Q}$-quaternion algebra ramified at $p$ and infinity.

I know that what I describe here is perhaps not the right way of proving the result you are asking, but it seems to me worth to mention.

In his '87 letter to Tate Serre proves:

${\rm Theorem:}$ Systems of mod $p$ Hecke eigenvalues arising from $M_k(\Gamma_1(N))$ are the same as those arising from locally constant function $f:G(A)\rightarrow\overline{F}_p$ that are left invariant under $G(\mathbf{Q})$ and right invariant under a certain open subgroup $K_N$.

Here $G(A)$ is the adelic group associated to $G$. Notice that the functions considered on the quaternion side are independent of the archimedean variable. Moreover, the double coset $G(\mathbf{Q})\backslash G(A)/K_N$ is finite and any mod $p$ system of eigenvalues arising from it can be lifted to characteristic zero.

Therefore applying the theorem and then lifting, we see that for any (char. zero) eigensystem $A=(a_\ell)$ arising from $M_k(\Gamma_1(N))$ there is a (char. zero) eigensystem $B=(b_\ell)$ arising from the space of locally constant functions $f:G(A)\rightarrow\mathbf{C}$ such that $A\equiv B$ mod $P$, where $P$ is a fixed prime of $\overline{\mathbf{Z}}$ lying over $p$.

Assuming that the automorphic form $\Pi_B$ on $G$ associated to $B$ is infinite dimensional, by the J-L correspondence we have that there is a cuspidal automorphic form $\Pi'_B$ on ${\rm GL}_2$ associated to the same eigensystem $B$. The type of $\Pi'_B$ at any finite place other than $p$ is the same as that of $\Pi_B$, while at infinity $\Pi'_B$ is the discrete series of lowest weight $2$. This basically says that there is a cusp form in $S_2(Np^n)$ whose associated system of eigenvalues is $B=(b_\ell)$.

We are only left with deciding when $\Pi_B$ is infinite dimensional, or can be chosen as such. This happens only for systems of eigenvalues of the form $B=(\chi(\ell)(1+\ell))_{\ell\nmid pN}$, where $\chi:\mathbf{Z}/p\rightarrow\mathbf{C}^*$ is any character (in order to show this one has to consider the particular shape of $K_N$, which I did not even define..). The reduction mod $P$ of such eigensystems are all of the form $(\ell^k+\ell^{k+1})_{\ell\nmid pN}$.

Concluding: Let $A=(a_\ell)$ be a sytstem of char. zero eigenvalues arising from $M_k(\Gamma_1(N))$, with $p\nmid N$. Assume that the mod $P$ reduction of $A$ is not of the form $(\ell^k+\ell^{k+1})_{\ell\nmid pN}$. Then, there exists a cusp form in $S_2(\Gamma_1(Np^n))$ such that its associated system of eigenvalues $B$ is congruent to $A$ mod $P$.