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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 non-transiting planet, TOI-1266 d (P = 32.5 d, $M_d$ = 3.68$^{+1.05}_{-1.11} M_{\oplus}$), and combine the TTVs with archival radial velocity (RV) measurements to improve our knowledge of the planetary masses and radii. We find that, consistent with previous studies, TOI-1266 b ($R_b$ = 2.52 $\pm$ 0.08 $R_{\oplus}$, $M_b$ = 4.46 $\pm$ 0.69 $M_{\oplus}$) has a low bulk density requiring the presence of a hydrogen-rich envelope, while TOI-1266 c ($R_c$ = 1.98 $\pm$ 0.10 $R_{\oplus}$, $M_c$ = 3.17 $\pm$ 0.76 $M_{\oplus}$) 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 non-resonant planet in between them. Our dynamical fits indicate that the inner and outer planet have significantly nonzero eccentricities ($e_b + e_d = 0.076^{+0.029}_{-0.019}$), 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., Comment: 36 pages, 19 figures, 5 tables, submitted to The Astronomical Journal