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The candidate supermassive black hole in the Galactic Centre, Sagittarius A* (Sgr A*), is known to be fed by a radiatively inefficient accretion flow (RIAF), inferred by its low accretion rate. Consequently, radiative cooling has in general been overlooked in the study of Sgr A*. However, the radiative properties of the plasma in RIAFs are poorly understood. In this work, using full 3D general-relativistic magneto-hydrodynamical simulations, we study the impact of radiative cooling on the dynamical evolution of the accreting plasma, presenting spectral energy distributions and synthetic sub-millimeter images generated from the accretion flow around Sgr A*. These simulations solve the approximated equations for radiative cooling processes self-consistently, including synchrotron, bremsstrahlung, and inverse Compton processes. We find that radiative cooling plays an increasingly important role in the dynamics of the accretion flow as the accretion rate increases: the mid-plane density grows and the infalling gas is less turbulent as cooling becomes stronger. The changes in the dynamical evolution become important when the accretion rate is larger than $10^{-8}\,M_{\odot}~{\rm yr}^{-1}$ ($\gtrsim 10^{-7} \dot{M}_{\rm Edd}$, where $\dot{M}_{\rm Edd}$ is the Eddington accretion rate). The resulting spectra in the cooled models also differ from those in the non-cooled models: the overall flux, including the peak values at the sub-mm and the far-UV, is slightly lower as a consequence of a decrease in the electron temperature. Our results suggest that radiative cooling should be carefully taken into account in modelling Sgr A* and other low-luminosity active galactic nuclei that have a mass accretion rate of $\dot{M} > 10^{-7}\,\dot{M}_{\rm Edd}$.