I have found out what's going on here and it is so trivial that I wonder why this did not occur to me earlier: The conclusion of the proof is that $K$ is the ray class field of $k$ modulo $N$ - but this means that $K$ is abelian over $k$, so the intermediate field $k(j_A, h(A_N))$ must be Galois over $k$ because all intermediate extensions of abelian extensions are Galois!

I have found out what's going on here and it is so trivial that I wonder why this did not occur to me earlier: The conclusion of the proof is that $K$ is the ray class field of $k$ modulo $N$ - but this means that $K$ is abelian over $k$, so the intermediate field $k(j_A, h(A_N))$ must be Galois over $k$ because all intermediate extensions of abelian extensions are Galois! Duh ...