According to Milne's Arithmetic Duality Theorems, Proposition I.6.4: $0 \to H^1(G_S, A[m]) \to H^1(K, A[m]) \to \oplus_{v \not\in S}H^1(K_v, A)$ for an abelian variety $A$ and a nonempty set of primes $S$ containing all infinite primes and the primes of bad reduction.

I want to prove this using étale cohomology. My idea was to extend $A$ to an abelian scheme and use the (excision) long exact sequence, but this lead me nowhere. Can someone give me some hints?