I'm not sure if I quite understood the question properly, but:
If a language $L$ is non-regular, then it might be context-free and in this case a contradiction can be provided as you explained for regular languages.
If a language is known to be non-CFL, then I would say there's no point of proving anymore that that language is not context-free. But if you want to do it again for any string $w$ in that language, at least the string must be able to be presented in the form $$w = uvwxy,$$ such that
- $|vwx| \leq l$, or the middle part of the string between $u$ and $y$ isn't very long
- $vx \neq \epsilon$, or at least another of the parts to be pumped $v$ and $x$ is not empty
- $uv^kwx^ky ∈ L$ for every $k ∈ \mathbb{N}$, or we can pump them in balance for an arbitrary amount $k$ and still belong in the language $L$.