Fixed target missing-momentum experiments such as LDMX and $M^3$ are powerful probes of light dark matter and other light, weakly coupled particles beyond the Standard Model (SM). Such experiments involve $∼10$ $GeV$ beam particles whose energy and momentum are individually measured before and after passing through a suitably thin target. If new states are radiatively produced in the target, the recoiling beam particle loses a large fraction of its initial momentum, and no SM particles are observed in a downstream veto detector. We explore how such experiments can use kinematic variables and experimental parameters, such as beam energy and polarization, to measure properties of the radiated particles and discriminate between models if a signal is discovered. In particular, the transverse momentum of recoiling particles is shown to be a powerful tool to measure the masses of new radiated states, offering significantly better discriminating ability compared to the recoil energy alone. We further illustrate how variations in beam energy, polarization, and lepton flavor (i.e., electron or muon) can be used to disentangle the possible the Lorentz structure of the new interactions.