Tractable $T$-matix model for reaction processes in muon catalyzed fusion $(dt\mu)_{J=v=0} \to \; \alpha + n + \mu + 17.6\, {\rm MeV} \; \mbox{or} \; (\alpha \mu)_{nl} + n +17.6 \,{\rm MeV}$

Reaction processes in muon catalyzed fusion ($\mu$CF), $(dt\mu)_{J=v=0} \to \alpha + n + \mu + 17.6\,{\rm MeV}\:$ or $ \;(\alpha \mu)_{nl} + n + 17.6\,{\rm MeV}$ in the D-T mixture was comprehensively studied by Kamimura, Kino and Yamashita [Phys. Rev. C 107, 034607 (2023)] by solving the $dt\mu$-$\alpha n\mu$ coupled channel (CC) Schr\"odinger equation under a boundary condition where the muonic molecule $(dt\mu)_{J=v=0}$ was set as the initial state and the outgoing wave was in the $\alpha n\mu$ channel. We approximate this CC framework and propose a considerably more tractable model using the $T$-matrix method based on the Lippmann-Schwinger equation. Nuclear interactions adopted in the $T$-matrix model are determined by reproducing the cross section of the reaction $d + t \to \alpha + n + 17.6\,{\rm MeV}$ at low energies. The cross section of the strong-coupling rearrangement reaction is presented in a simple closed form based on our new model. This $T$-matrix model have reproduced most of the calculated results on the above $\mu$CF reaction reported by Kamimura et al. (2023) and is applicable to other $\mu$CF systems such as $(dd\mu)$, $(tt\mu)$, $(dt\mu)^*$, $(dd\mu)^*$.