@YemonChoi We can think of $\widehat G$ as living inside $\mathfrak{t}^\vee$, where $\mathfrak{t}=\mathrm{Lie}(T)$ is the Lie algebra of the maximal torus. We can order the sum $\sum_\pi a_\pi tr(\pi)$ by looking at those $\pi$ which live in increasingly large balls (any choice of metric on $\mathfrak{t}^\vee$ will do).

Christian—your answer is correct. If you make it into an official answer I'll accept it.

Thanks Christian - I'll see if I can prove it for bounded + a.e. continuous.