Getting from one language to the other using closure properties(automata)
If I understand your argument correctly, you are reducing languages in the wrong direction.


  If $L$ is not context-free, then $K$ is not context-free.


Is equivalent to


  If $K$ is context-free, then $L$ is context-free.


We have to reverse the construction, as we are using the closure properties of the context-free languages. In do not know of any useful closure properties of the non-context-free languages.

PS. The pumping argument seems out of place here.