Average atom (A-atom) potentials provide a mean-field description of a chemically disordered alloy and are used to predict the properties of solid solutions without short-range order. Such potentials are usually averaged from an existing interatomic potential and are therefore only as accurate as the parent model. Accurate interatomic potentials are themselves difficult to parameterize and can require large training datasets. Here we benchmark a recently developed formalism that computes an exact A-atom potential directly from a linear atomic cluster expansion (ACE). We first fit a linear ACE to Fe-W data generated with an embedded atom method (EAM) potential. The resulting A-atom potential reproduces the properties of the disordered phase computed from an explicit random supercell and from a conventional A-atom potential averaged from the same EAM potential. We then fit a linear ACE to energies and forces computed from electronic structure calculations for about 1500 small Mo-Nb structures with an average of 7 atoms per structure. The A-atom potential derived from this ACE reproduces the DFT elastic constants, lattice parameter, mixing enthalpy, and Bain path of special quasirandom structures. Of the three chemical site bases, only the occupation basis also reproduces the DFT surface energies of the alloy. For Mo-Nb, the A-atom potential also predicts that the ideal solution entropy outweighs the destabilizing vibrational contribution to the finite-temperature free energy of the disordered phase. These benchmarks show that the properties of disordered alloys can be recovered from small training datasets when the chemical site basis is chosen carefully.