This might be addressed in Lang's book "Topics in Galois cohomology" near the end (where he does discuss Tate's theorem on abelian varieties that you mention).  But here is a proof when $L/K$ is separable (which you might regard as a sketch, but does give all the main ideas).  

First, by the relationship between cup products and connecting homomorphisms (and the identification of $A^t(K)$ with ${\rm{Ext}}^1_K(A, \mathbf{G}_m)$ *functorially* in $A$) we see that the pairing $$A^t(K) \times {\rm{H}}^1(K,A) \rightarrow {\rm{H}}^2(K, \mathbf{G}_m) = \mathbf{Q}/\mathbf{Z}$$ 
identifies covariant functoriality in degree-1 cohomology as adjoint to dual-functoriality of abelian varieties.  

Let $B$ denote the Weil restriction of scalars ${\rm{R}}_{L/K}(A_L)$; this is an abelian variety precisely because $L/K$ is separable. (If $L/K$ is not separable then $B$ is a smooth connected commutative $K$-group of dimension $[L:K]\dim(A)$ but is always non-proper if $A \ne 0$.) Let $j:A \rightarrow B$ be the natural inclusion.  Ultimately we are going to transform your question into the above functoriality of the Tate pairing over $K$ applied to the $K$-homomorphism $j$.

 By Shapiro's Lemma considerations, we naturally identify ${\rm{H}}^i(L,A)$ with ${\rm{H}}^i(K,B)$, and (check!) this identifies the restriction map on ${\rm{H}}^1$'s with ${\rm{H}}^1(j)$.  Likewise, by the compatibility of Weil restriction with the formation of dual abelian variety (using the "norm" of the Weil restriction of the Poincare bundle), the norm map $A^t(L)\rightarrow A^t(K)$ is identified with the map on $K$-points induced by the dual homomorphism $j^t:B^t \rightarrow A^t$.  Also, and most crucially, by a bit of diagram chasing (using the role of "norm of Poincare bundle" above) we see that the Tate pairing for $B$ over $K$ is identified with the composition of the Tate pairing for $A_L$ over $L$ and the "norm" map on Brauer groups $${\rm{Br}}(L) = {\rm{H}}^2(L, \mathbf{G}_m) =
{\rm{H}}^2(K, {\rm{R}}_{L/K}(\mathbf{G}_m))\rightarrow {\rm{H}}^2(K,\mathbf{G}_m)={\rm{Br}}(K).$$
But when these flanking Brauer groups are identified with $\mathbf{Q}/\mathbf{Z}$, this composite map is the *identity*, as we see by analyzing pre-composition with the surjective restriction ${\rm{Br}}(K) \rightarrow {\rm{Br}}(L)$ (that intertwines with $[L:K]$ on $\mathbf{Q}/\mathbf{Z}$ via local class field theory, and the composition of $\mathbf{G}_m \rightarrow {\rm{R}}_{L/K}(\mathbf{G}_m)$ with the "norm" map ${\rm{R}}_{L/K}(\mathbf{G}_m) \rightarrow \mathbf{G}_m$ is $t \mapsto t^{[L:K]}$).

So putting it all together, the diagram you want to commute really does translate into the elementary functoriality of the Tate pairing over $K$, applied to the map $j$ between abelian varieties over $K$.