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Over any field $k$, $\hat A=Ext(A,G_m)$ in the abelian category (see "Is the category of commutative group schemes abelian" here on MO) of commutative group schemes of finite type over $k$. There is a natural isomorphism $Ext(A \times A,G_m) \cong Ext(A,G_m) \oplus Ext(A,G_m)$ ($Ext$ is a bi-additive functor), from which a natural isomorphism $\widehat{A \times A} \rightarrow \hat A \times \hat A$ that you seek . The Poincare bundles on $(A \times A) \times (\hat A \times \hat A)$ and on $\widehat{A \times A} \times (A \times A)$ should be easy to relate as well -- just pullbacks via the canonical isomorphisms mentioned above.