TOI-1266 is a benchmark system of two temperate (<450 K) sub-Neptune-sized planets orbiting a nearby M dwarf exhibiting a rare inverted architecture with a larger interior planet. In this study, we characterize transit timing variations (TTVs) in the TOI-1266 system using high-precision ground-based follow-up and new TESS data. We confirm the presence of a third exterior nontransiting planet, TOI-1266 d (P = 32.5 days, Md = 3.68 ), and combine the TTVs with archival radial velocity measurements to improve our knowledge of the planetary masses and radii. We find that, consistent with previous studies, TOI-1266 b (Rb = 2.52 ± 0.08 R⊕, Mb = 4.46 ± 0.69 M⊕) has a low bulk density requiring the presence of a hydrogen-rich envelope, while TOI-1266 c (Rc = 1.98 ± 0.10 R⊕, Mc = 3.17 ± 0.76 M⊕) has a higher bulk density that can be matched by either a hydrogen-rich or water-rich envelope. Our new dynamical model reveals that this system is arranged in a rare configuration with the inner and outer planets located near the 3:1 period ratio with a nonresonant planet in between them. Our dynamical fits indicate that the inner and outer planet have significantly nonzero eccentricities ( ), suggesting that TOI-1266 b may have an inflated envelope due to tidal heating. Finally, we explore the corresponding implications for the formation and long-term evolution of the system, which contains two of the most favorable cool (<500 K) sub-Neptunes for atmospheric characterization with JWST.