We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.