  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.
