If you require that the matrix $C$ in \eqref{2} preserves the involutions \eqref{3}, then you get the consequence $\tilde{\gamma}^\mu = C C^* \tilde{\gamma}^\mu (C C^*)^{-1}$, which then implies that $C C^* = u$ is some unit in the center of the Clifford algebra such that $u = u^*$. Equivalently $C^* = u C^{-1}$. Rescaling $C=v D$, where $v$ is another scalar unit (which obviously cancels in the formula $C \gamma C^{-1} = D \gamma D^{-1}$), we get $D^* = \frac{u}{v^* v} D^{-1}$. So we can normalize the involution properties of $C$ by choosing $u/(v^* v)$ to be some canonical value.
The center of a complex Clifford algebra is either 1- or 2-dimensional, in a pattern that depends on the number of generators (which I'm always too lazy to look up). But supposing the center is 1-dimensional, that is consisting of scalar matrices $\mathbb{C} I$, for any $u$ we can choose $v$ such that $u/(v^* v) = \pm 1$. In that case, if you can find one tuple of $\gamma$-matrices satisfying \eqref{3}, the rest are classified by \eqref{2} where it is sufficient to take $C^* = \pm C^{-1}$. The case of the 2-dimensional center can be left as an exercise.
