Interatomic potentials are central tools in the atomistic modeling of materials. The atomic cluster expansion (ACE) parameterizes such potentials from ab initio data, conventionally encoding the chemical degrees of freedom with a one-hot representation that yields chemically stratified models. The alternative chemical representations used in on-lattice configurational cluster expansions have not been assessed for interatomic potentials. Here we revisit the multicomponent ACE for an arbitrary chemical site basis. We then establish an exact analytical mapping between a fitted linear ACE and the average-atom potential that describes a perfectly random alloy. We benchmark potentials built on the occupational, Chebyshev, and conventional ACE bases against solute binding and vacancy formation energies in Mg-Nd, and against the mixing enthalpies of the Mo-Nb and Cr-W solid solutions. When training data are scarce, the occupational basis converges fastest and offers the best control over targeted material properties, while the conventional and Chebyshev bases face challenges in reproducing these properties. The occupational basis likewise yields the most reliable average-atom description of disordered alloy thermodynamics. In the large-data limit the three bases perform identically. The chemical basis is therefore a design choice that governs data efficiency. Its explicit treatment opens a route to average-atom potentials for the thermodynamic, mechanical, and kinetic properties of concentrated alloys.