Hexagonal perovskite 15R-BaMnO2.99 with a ratio of cubic to hexagonal layers of 1/5 in the unit cell is an antiferromagnetic insulator that orders at a Neel temperature TN = 220 K. Here we report structural, magnetic, dielectric and thermal properties of single crystal BaMnO2.99 and its derivatives BaMn0.97Li0.03O3 and Ba0.97K0.03MnO3. The central findings of this work are: (1) these materials possess a usually large, high-temperature magnetoelectric effect that amplifies the dielectric constant by more than an order of magnitude near their respective Neel temperature; (2) Li and K doping can readily vary the ratio of cubic to hexagonal layers and cause drastic changes in dielectric and magnetic properties; in particular, a mere 3% Li substitution for Mn significantly weakens the magnetic anisotropy and relaxes the lattice; consequently, the dielectric constant for both the a- and c-axis sharply rises to 2500 near the Neel temperature. This lattice softening is also accompanied by weak polarization. These findings provide a new paradigm for developing novel, high-temperature magnetoelectric materials that may eventually contribute to technology.