How to prove a norm identity for a Banach space and its dual

You don't need completeness of $X$. This is true in normed spaces. Your last idea is a good one:

We may assume $x \neq 0$. Define the functional

$$\varphi: \Bbb{C}x \to \Bbb{C}: \lambda x \mapsto \lambda \Vert x \Vert$$

Then it is easily checked that $\Vert \varphi \Vert =1$ (the inequality $\leq$ is obvious, and then note that $\varphi(x/\Vert x \Vert) = 1$ so also $\Vert \varphi\Vert \geq 1$). By Hahn-Banach, we can extend to a functional $\tilde{\varphi}: X \to \Bbb{C}$ with $\Vert \tilde{\varphi} \Vert =1$ and this is the functional you are looking for.