We study neutrino-induced charged-current (CC) pi(0) production on carbon nuclei using events with fully imaged final-state proton-pi(0) systems. Novel use of final-state correlations based on transverse kinematic imbalance enables the first measurements of the struck nucleon's Fermi motion, of the intranuclear momentum transfer (IMT) dynamics, and of the final-state hadronic momentum configuration in neutrino pion production. Event distributions are presented for (i) the momenta of neutrino-struck neutrons below the Fermi surface, (ii) the direction of missing transverse momentum characterizing the strength of IMT, and (iii) proton-pion momentum imbalance with respect to the lepton scattering plane. The observed Fermi motion and IMT strength are compared to the previous MINER.A measurement of neutrino CC quasielastic-like production. The measured shapes and absolute rates of these distributions, as well as the cross section asymmetries, show tensions with predictions from current neutrino generator models.