The KATRIN experiment is designed for a direct and model-independent determination of the effective electron anti-neutrino mass via a high-precision measurement of the tritium $beta$-decay endpoint region with a sensitivity on $m_ u$ of 0.2$,$eV/c$^2$ (90% CL). For this purpose, the $beta$-electrons from a high-luminosity windowless gaseous tritium source traversing an electrostatic retarding spectrometer are counted to obtain an integral spectrum around the endpoint energy of 18.6$,$keV. A dominant systematic effect of the response of the experimental setup is the energy loss of $beta$-electrons from elastic and inelastic scattering off tritium molecules within the source. We determined the linebreak energy-loss function in-situ with a pulsed angular-selective and monoenergetic photoelectron source at various tritium-source densities. The data was recorded in integral and differential modes; the latter was achieved by using a novel time-of-flight technique. We developed a semi-empirical parametrization for the energy-loss function for the scattering of 18.6-keV electrons from hydrogen isotopologs. This model was fit to measurement data with a 95% T$_2$ gas mixture at 30$,$K, as used in the first KATRIN neutrino mass analyses, as well as a D$_2$ gas mixture of 96% purity used in KATRIN commissioning runs. The achieved precision on the energy-loss function has abated the corresponding uncertainty of $sigma(m_ u^2)<10^{-2},mathrm{eV}^2$ [arXiv:2101.05253] in the KATRIN neutrino-mass measurement to a subdominant level.
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