We study $q$-pushTASEP, a discrete time interacting particle system whose distribution is related to the $q$-Whittaker measure. We prove a uniform in $N$ lower tail bound on the fluctuation scale for the location $x_N(N)$ of the right-most particle at time $N$ when started from step initial condition. Our argument relies on a map from the $q$-Whittaker measure to a model of periodic last passage percolation (LPP) with geometric weights in an infinite strip that was recently established in [arXiv:2106.11922]. By a path routing argument we bound the passage time in the periodic environment in terms of an infinite sum of independent passage times for standard LPP on $N\times N$ squares with geometric weights whose parameters decay geometrically. To prove our tail bound result we combine this reduction with a concentration inequality, and a crucial new technical result -- lower tail bounds on $N\times N$ last passage times uniformly over all $N \in \mathbb N$ and all the geometric parameters in $(0,1)$. This technical result uses Widom's trick [arXiv:math/0108008] and an adaptation of an idea of Ledoux introduced for the GUE [Led05a] to reduce the uniform lower tail bound to uniform asymptotics for very high moments, up to order $N$, of the Meixner ensemble. This we accomplish by first obtaining sharp uniform estimates for factorial moments of the Meixner ensemble from an explicit combinatorial formula of Ledoux [Led05b], and translating them to polynomial bounds via a further careful analysis and delicate cancellation.