Robustness of non-context-free proof against trivial manipulation
Note $\left\{x^{\mathrm{R}}x\mid x\in\Sigma^*\right\}$ is context free, where $x^{\mathrm{R}}$ means the reverse of $x$. 


  Or even worse, there may be some simple manipulation that push $L$ down to CFL again.


So if you consider reversing as a simple manipulation, the answer is indeed yes.