The ultrafilter $U$ you get is not iterable with $L_\alpha$, since the power set of $\kappa$ in $L_\alpha$ and the target $L_\beta$ is not the same. So there is no way to define the next step of the iteration. What you need to iterate the ultrafilter is also weak amenability, which is equivalent to saying that the ultrapower has the same subsets of $\kappa$. 

Meanwhile, the weak amenability hypothesis is part of Kanamori's framework in chapter 19, and so there is no contradiction.