As in my comment, Theorem 2 of van Dijk - Computation of certain induced characters of $p$-adic groups (MR) says, with the notation of that result, that \begin{align*} \Theta_{\mathrm{ind}_P^G(\rho\chi)}(f) & = \int_M \chi(m)\int_K \int_N f(k m n k^{-1})\delta_P(m)^{1/2}\theta_\rho(m)\mathrm dn\,\mathrm dk\,\mathrm dm \\ & = \sum_{m \in M/K \cap M} \chi(m)\operatorname{meas}(K \cap M)\int_K \int_N f(k m n k^{-1})\delta_P(m)^{1/2}\theta_\rho(m)\mathrm dn\,\mathrm dk. \end{align*} (Actually the stated integral is in a different order, but each integral is compactly supported, so the interchange is OK.) The sum is finite because $f$ is compactly supported.

As in my comment, Theorem 2 of van Dijk - Computation of certain induced characters of $p$-adic groups (MR) says, with the notation of that result, that \begin{align*} \Theta_{\mathrm{ind}_P^G(\rho\chi)}(f) & = \int_M \chi(m)\int_K \int_N f(k m n k^{-1})\delta_P(m)^{1/2}\theta_\rho(m)\mathrm dn\,\mathrm dk\,\mathrm dm \\ & = \sum_{m \in M/K \cap M} \chi(m)\operatorname{meas}(K \cap M)\int_K \int_N f(k m n k^{-1})\delta_P(m)^{1/2}\theta_\rho(m)\mathrm dn\,\mathrm dk. \end{align*} (Actually the stated integral is in a different order, but each integral is compactly supported, so the interchange is OK.) The sum is finite because $f$ is compactly supported.