We study future coherent elastic neutrino-nucleus scattering (CE$ u$NS) modifications from a variety of possible models at the Coherent CAPTAIN Mills (CCM) experiment at Los Alamos. We show that large regions of Non-Standard Neutrino Interaction (NSI) parameter space will be excluded rapidly, and that stringent new bounds on the gauge coupling in $Z$ models will also be placed. As a result, CCM will be able to rule out LMA-D solutions for a large class of models with MeV-scale mediators.