While interaction with circumstellar material is known to play an important role in Type IIn supernovae (SNe), analyses of the more common SNe IIP and IIL have not traditionally included interaction as a significant power source. However, recent campaigns to observe SNe within days of explosion have revealed narrow emission lines of high-ionization species in the earliest spectra of luminous SNe II of all subclasses. These "flash ionization" features indicate the presence of a confined shell of material around the progenitor star. Here we present the first low-luminosity (LL) SN to show flash ionization features, SN 2016bkv. This SN peaked at MV = -16 mag and has H{alpha} expansion velocities under 1350 km s-1 around maximum light, placing it at the faint/slow end of the distribution of SNe IIP (similar to SN 2005cs). The light-curve shape of SN 2016bkv is also extreme among SNe IIP. A very strong initial peak could indicate additional luminosity from circumstellar interaction. A very small fall from the plateau to the nickel tail indicates unusually large production of radioactive nickel compared to other LL SNe IIP. A comparison between nebular spectra of SN 2016bkv and models raises the possibility that SN 2016bkv is an electron-capture supernova.