In a previous work \cite{Buttazzo:2020ibd}, it has been pointed out that heavy new physics (NP) contributions to lepton dipole moments and high-energy cross-sections of lepton pairs into Higgs and $\gamma/Z$ bosons are connected, if the electroweak symmetry breaking is realized linearly. As a consequence, a multi-TeV muon collider would provide a unique test of NP in the muon $g$-2 through the study of high-energy processes such as $\mu^+ \mu^-\to h +\gamma /Z$. Since the analysis involved a SMEFT approach to Higgs-processes, it could also be studied by the more general HEFT formulation. We compute the modification of the high-energy cross-section $\mu^+ \mu^-\to h +\gamma/Z$ and $h\to \mu^{+} \mu^{-} +\gamma /Z$ decay using the dimension six HEFT Lagrangian and compare them with the SMEFT analysis. We find that, in some reasonable cases, the HEFT approach could lead to a higher sensitivity to test the NP contributions in the muon $g$-2, requiring thus a less energetic muon collider for such a purpose.