  CaCo$_{2-y}$As$_2$ is an unusual itinerant magnet with signatures of extreme
magnetic frustration. The conditions for establishing magnetic order in such
itinerant frustrated magnets, either by reducing frustration or enhancing the
Stoner parameter, is an open question. Here we use results from inelastic
neutron scattering and magnetic susceptibility measurements and density
functional theory calculations to show that hole doping in
Ca(Co$_{1-x}$Fe$_{x}$)$_{2-y}$As$_{2}$ suppresses magnetic order by quenching
the associated magnetic moment while maintaining the same level of magnetic
frustration. The suppression is due to tuning the Fermi energy away from a peak
in the electronic density of states originating from a flat conduction band.
This results in the complete elimination of the magnetic moment by
$x\approx0.25$, providing a clear example of a Stoner-like transition.
