Shot-noise measurements with a quantum point contact provide a direct probe of the quasiparticle charge in Laughlin fractional quantum Hall (FQH) states. However, the interpretation of such measurements becomes more subtle for multichannel edges, where the point-contact tunneling noise coexists with noise generated by intermode equilibration and heating. We develop a general framework that incorporates both these contributions and apply it to the paradigmatic ν=2/3 FQH edge. We derive expressions for the drain auto- and cross-correlations in a four-terminal QPC geometry and show that equilibration-induced heating affects the auto- and cross-correlations in parametrically different way than the intrinsic partition noise. This distinction permits us to propose a specific combination of the two correlations which strongly suppresses the heating contribution and fully recovers the tunneling quasiparticle charge. Our results thereby provide a systematic framework for interpreting shot-noise measurements in complex FQH edges and can be further extended to more general counterpropagating edge structures.