We investigate non-stationary heat transfer in the Kipnis-Marchioro-Presutti (KMP) lattice gas model at long times in one dimension when starting from a localized heat distribution. At large scales this initial condition can be described as a delta-function, $u(x,t=0)=W δ(x)$. We characterize the process by the heat, transferred to the right of a specified point $x=X$ by time $T$, $$ J=\int_X^\infty u(x,t=T)\,dx\,, $$ and study the full probability distribution $\mathcal{P}(J,X,T)$. The particular case of $X=0$ has been recently solved [Bettelheim \textit{et al}. Phys. Rev. Lett. \textbf{128}, 130602 (2022)]. At fixed $J$, the distribution $\mathcal{P}$ as a function of $X$ and $T$ has the same long-time dynamical scaling properties as the position of a tracer in a single-file diffusion. Here we evaluate $\mathcal{P}(J,X,T)$ by exploiting the recently uncovered complete integrability of the equations of the macroscopic fluctuation theory (MFT) for the KMP model and using the Zakharov-Shabat inverse scattering method. We also discuss asymptotics of $\mathcal{P}(J,X,T)$ which we extract from the exact solution, and also obtain by applying two different perturbation methods directly to the MFT equations.