Can an alphabet be extended in a reduction proof? (with sample problem)
A (computable) reduction from problem P to problem Q is a (computable) function that takes an instance of P to an instance of Q. In your case $P=\mathcal{E}^c$ and $Q = L^c$. So your reduction is "legal" (that is, conforms to the definition of reduction) if it takes an instance of $\mathcal{E}^c$ to an instance of $L^c$. That's all a function has to do in order to be a reduction.

(In principle there is no such thing as an instance of a problem – every string over the appropriate alphabet is an instance. But let's ignore this small issue.)