The wide gap perovskite semiconductor ${\mathrm{BaSnO}}_{3}$ has attracted much interest since the discovery of room temperature electron mobility up to $320\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{2}\phantom{\rule{0.16em}{0ex}}{\mathrm{V}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$ in bulk crystals. Motivated by applications in oxide heterostructures, rapid progress has been made with ${\mathrm{BaSnO}}_{3}$ films, although questions remain regarding transport mechanisms and mobility optimization. Here we report on a detailed study of epitaxial ${\mathrm{BaSnO}}_{3}$ electric double layer transistors based on ion gel electrolytes, enabling wide-doping-range studies of transport in single films. The work spans an order of magnitude in initial $n$ doping ($\ensuremath{\sim}{10}^{19}$ to ${10}^{20}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}3}$, with both oxygen vacancies and La), film thicknesses from 10--50 nm, and measurements of resistance, Hall effect, mobility, and magnetoresistance. In contrast with many oxides, electrolyte gating of ${\mathrm{BaSnO}}_{3}$ is found to be essentially reversible over an exceptional gate voltage window (approaching $\ifmmode\pm\else\textpm\fi{}4\phantom{\rule{0.16em}{0ex}}\mathrm{V}$), even at 300 K, supported by negligible structural modification in operando synchrotron x-ray diffraction. We propose that this occurs due to a special situation in ${\mathrm{BaSnO}}_{3}$, where electrochemical gating via oxygen vacancies is severely limited by their low diffusivity. Wide-range reversible modulation of transport is thus achieved (in both electron accumulation and depletion modes), spanning strongly localized, weakly localized, and metallic regimes. Two-channel conduction analysis is then combined with self-consistent Schr\"odinger-Poisson and Thomas-Fermi modeling to extract accumulation layer electron densities and mobilities. Electrostatic electron densities approaching ${10}^{14}\phantom{\rule{0.16em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}2}$ are shown to increase room temperature mobility by up to a factor of $\ensuremath{\sim}24$. These results lay the groundwork for future studies of electron-density-dependent phenomena in high mobility ${\mathrm{BaSnO}}_{3}$, and significantly elucidate oxide electrolyte gating mechanisms.