We report the discovery and characterization of a small planet, TOI-1408 c, on a 2.2 day orbit located interior to a previously known hot Jupiter, TOI-1408 b (P = 4.42 days, M = 1.86 ± 0.02 M Jup, R = 2.4 ± 0.5 R Jup) that exhibits grazing transits. The two planets are near 2:1 period commensurability, resulting in significant transit timing variations (TTVs) for both planets and transit duration variations for the inner planet. The TTV amplitude for TOI-1408 c is 15% of the planet's orbital period, marking the largest TTV amplitude relative to the orbital period measured to date. Photodynamical modeling of ground-based radial velocity (RV) observations and transit light curves obtained with the Transiting Exoplanet Survey Satellite and ground-based facilities leads to an inner planet radius of 2.22 ± 0.06 R ⊕ and mass of 7.6 ± 0.2 M ⊕ that locates the planet into the sub-Neptune regime. The proximity to the 2:1 period commensurability leads to the libration of the resonant argument of the inner planet. The RV measurements support the existence of a third body with an orbital period of several thousand days. This discovery places the system among the rare systems featuring a hot Jupiter accompanied by an inner low-mass planet.