One way to see this is to note that $T$ splits from any locally compact abelian group (D.L. Armacost, The Structure of Locally Compact Abelian Groups, 6.16). If the cocycle is commutative, then the associated extension $$ 0 \to T \to E \to G\to 0 $$ is a short exact sequence of abelian groups. Since $T$ splits, the cocycle is a coboundary.
This is discussed in my paper "Locally compact Abelian groups with symplectic self-duality"; see http://www.imsc.res.in/~amri/summaries.html#ssdg in Section 3.