The solar heating problem involves a coupled system (including the photosphere and chromosphere), thus requiring its study as a whole system. This requirement often leads to the combination of observations with different instrumental effects. One important missing piece in this context is the capability to observationally constrain the various physical mechanisms that heat the outer layers as predicted from numerical simulations. In this contribution we present the implementation of a newly-developed inversion algorithm that expands the capabilities of the NLTE inversion code STiC to better handle inversions combining multi-resolution observations, which can provide more constrained physical inferences. After we have checked its performance, we apply this inversion method to a quiet-Sun area co-observed with the SST (CRISP and CHROMIS) and IRIS spectrograph. After inferring the various physical properties of the atmosphere we compute the radiative losses from different sources and discuss their potential physical origin.