I'm afraid that this question has a disappointingly simple answer. Yes, the values of the Iwahori-spherical Whittaker functions have an interpretation as characters of representations; but they are one-dimensional representations!
To see this, it suffices to check out $n = 2$. Here the Satake parameter is an unordered pair of (nonzero) complex numbers $\alpha, \beta$. One checks that $\mathfrak{W}(\tau, \psi)^{J_2}$ is 2-dimensional with a natural basis $$\{ \mathcal{W}_\alpha, \mathcal{W}_\beta \} $$and the values of these on the maximal torus are given (up to some powers of $q$ which I am ignoring) by$$\mathcal{W}_\alpha \left( \begin{pmatrix} \pi^r & 0 \\ 0 & \pi^s\end{pmatrix}\right) = \alpha^r \beta^s. $$
The picture for general $n$ is similar: there is a natural basis of the Whittaker functions indexed by the $n!$ Weyl-group permutations $w \cdot \tau$ of the inducing character $\tau$, and the Whittaker function associated to $w \cdot \tau$ just restricts to $w \cdot \tau$ on the torus.
So there is much less interesting structure in the values of the Iwahori-level Whittaker functions than there is in the spherical ones. The interesting combinatorics is in how to stick together Iwahori Whittaker functions to get spherical Whittaker functions.