Better not to prove it, as it is wrong. Writing $M[p] = \lbrace m \in M: pm = 0 \rbrace$ for the exact $p$-torsion of a module $M$, I understand your question as follows: If the exact sequence $$0 \rightarrow A[p] \rightarrow A \rightarrow A/A[p] \rightarrow 0$$ remains exact modulo $p$ (which is not always the case: it is equivalent to $A[p] \cap pA = 0$), then this exact reduced sequence $$0 \rightarrow (A[p])/p \rightarrow A/p \rightarrow (A/A[p])/p \rightarrow 0$$ splits. A counterexample is $A = \mathbb{Z}_p[\![X]\!] /(pX)$ where the reduced sequence $$ 0 \rightarrow X\mathbb{F}_p[\![X]\!] \rightarrow \mathbb{F}_p[\![X]\!] \rightarrow \mathbb{F}_p \rightarrow 0$$ is exact, but does not split, nor is there even any chance to write the middle term as direct sum of the outer ones (as $\mathbb{F}_p[\![X]\!]$-module).
Maybe the illusion that the statement might be true comes from the fact that it is true for $p$-torsion modules (in particular, pseudonull modules) and the standard modules $\bigoplus_i\mathbb{Z}_p[\![X]\!]/(p^{\mu_i})\oplus\bigoplus_j\mathbb{Z}_p[\![X]\!]/(f_j^{m_j})$, as a direct check shows.
Computing a pseudo-isomorphism from the above $A$ to the standard module $\mathbb{Z}_p[\![X]\!]/(p)\oplus\mathbb{Z}_p[\![X]\!]/(X)$ and seeing where things go wrong might be a good exercise in Iwasawa theory, compatible to questions like Dual of a module , Iwasawa algebra , Rank of a $ \mathbb{Z}_{p}[[T]] $ module or Iwasawa invariants.