/ AbstractDMRadio-m$^3$ is an experiment that is designed to be sensitive to KSVZ and DFSZ QCD axion models in the 10--200\,MHz (41 neV$/c^2$ -- 0.83 $μ$eV/$c^2$) range. The experiment uses a solenoidal dc magnetic field to convert an axion dark-matter signal to an ac electromagnetic response in a coaxial copper pickup. The current induced by this axion signal is measured by dc SQUIDs. In this work, we present the electromagnetic modeling of the response of the experiment to an axion signal over the full frequency range of DMRadio-m$^3$, which extends from the low-frequency, lumped-element limit to a regime where the axion Compton wavelength is only a factor of two larger than the detector size. With these results, we determine the live time and sensitivity of the experiment. The primary science goal of sensitivity to DFSZ axions across 30--200 MHz can be achieved with a $3σ$ live scan time of 2.9 years.