Can any c.e. language with infinite words be decomposed into infinite CFLs with infinite words?
No, we get counterexamples by considering resource-bounded randomness. In fact the gap between c.e. and $\mathsf{CFL}$ is wide.

Let $R$ be exponential-time random. Then $R$ is $\mathsf{NP}$-immune, i.e., it has no infinite subset in $\mathsf{NP}$. In particular it has no infinite subset that is a context-free language.