The Sloan Digital Sky Survey-IV/Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) Survey data provide an unprecedented opportunity to study the internal motions of galaxies and, in particular, represent the largest sample of barred galaxy kinematic maps obtained to date. We present results from Nirvana, our nonaxisymmetric kinematic modeling code built with a physically motivated Bayesian forward modeling approach, which decomposes MaNGA velocity fields into first- and second-order radial and tangential rotational modes in a generalized and minimally supervised fashion. We use Nirvana to produce models and rotation curves for 1263 unique barred MaNGA galaxies and a matched unbarred control sample. We present our modeling approach, tests of its efficacy, and validation against existing visual bar classifications. Nirvana finds elevated noncircular motions in galaxies identified as bars in imaging, and bar position angles that agree well with visual measurements. The Nirvana-MaNGA barred and control samples provide a new opportunity for studying the influence of nonaxisymmetric internal disk kinematics in a large statistical sample.