We present near-UV (NUV) flare spectra from the Hubble Space Telescope (HST)/Cosmic Origins Spectrograph during two moderate-amplitude U-band flares on the dM4e star GJ 1243. These spectra are some of the first accurately flux-calibrated, NUV flare spectra obtained over the impulsive phase in M dwarf flares. We observed these flares with a fleet of nine ground-based telescopes simultaneously, which provided broadband photometry and low-resolution spectra at the Balmer jump. An increase in the broadband continuum occurred with a signal-to-noise ratio >20 in the HST spectra, while numerous Fe II lines and the Mg II lines also increased but with smaller flux enhancements than the continuum radiation. These two events produced the most prominent Balmer line radiation and the largest Balmer jumps that have been observed to date in dMe flare spectra. A T = 9000 K blackbody underestimates the NUV continuum flare flux by a factor of two and is a poor approximation to the white light in these types of flare events. Instead, our data suggest that the peak of the specific continuum flux density is constrained to U-band wavelengths near the Balmer series limit. A radiative-hydrodynamic simulation of a very high energy deposition rate averaged over times of impulsive heating and cooling better explains the properties of the λ > 2500 Å flare continuum. These two events sample only one end of the empirical color–color distribution for dMe flares, and more time-resolved flare spectra in the NUV, U band, and optical from 2000 to 4200 Å are needed during more impulsive and/or more energetic flares.