We present the results of a medium resolution optical spectroscopic survey of 92 cool ($3,000 lesssim T_{rm eff} lesssim 4,500,$K) southern TESS candidate planet hosts, and describe our spectral fitting methodology used to recover stellar parameters. We quantify model deficiencies at predicting optical fluxes, and while our technique works well for $T_{rm eff}$, further improvements are needed for [Fe/H]. To this end, we developed an updated photometric [Fe/H] calibration for isolated main sequence stars built upon a calibration sample of 69 cool dwarfs in binary systems, precise to $pm0.19,$dex, from super-solar to metal poor, over $1.51 < {rm Gaia}~(B_P-R_P) < 3.3$. Our fitted $T_{rm eff}$ and $R_star$ have median precisions of 0.8% and 1.7%, respectively and are consistent with our sample of standard stars. We use these to model the transit light curves and determine exoplanet radii for 100 candidate planets to 3.5% precision and see evidence that the planet-radius gap is also present for cool dwarfs. Our results are consistent with the sample of confirmed TESS planets, with this survey representing one of the largest uniform analyses of cool TESS candidate planet hosts to date.
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