Thin garnet films are becoming central for magnon-spintronics and spin-orbitronics devices as they show versatile magnetic properties together with low magnetic losses. These fields would benefit from materials in which heat does not affect the magnetization dynamics, an effect known as the non-linear thermal frequency shift. In this study, low damping Bi substituted Iron garnet (Bi:YIG) ultra-thin films have been grown using Pulsed Laser Deposition. Through a fine tuning of the growth parameters, the precise control of the perpendicular magnetic anisotropy allows to achieve a full compensation of the dipolar magnetic anisotropy. Strikingly, once the growth conditions are optimized, varying the growth temperature from 405 °C to 475 °C as the only tuning parameter induces the easy-axis to go from out-of-plane to in-plane. For films that are close to the dipolar compensation, Ferromagnetic Resonance measurements yield an effective magnetization µ$_0$M$_{eff}$ (T) that has almost no temperature dependence over a large temperature range (260 K to 400 K) resulting in an anisotropy temperature exponent of 2. These findings put Bi:YIG system among the very few materials in which the temperature dependence of the magnetic anisotropy varies at the same rate than the saturation magnetization. This interesting behavior is ascribed phenomenologically to the sizable orbital moment of Bi$^{3+}$