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The Josephson plasma resonance (JPR) in layered high temperature superconducting cuprates provides a useful probe of the superconducting condensate, as well as an avenue to couple incident electromagnetic fields to the condensate. We enhance this light-matter coupling by applying interchangeable metamaterial tapes to a c-axis single crystal of La_2-xSr_xCuO₄ and perform linear THz time domain spectroscopy in reflection in order to identify anti-crossing behavior characteristic of strong coupling. We then drive the hybrid superconducting metamaterial system with large single cycle THz fields in excess of 100kV/cm in order to explore emergent nonlinear behavior of the coupled system. This work informs future possibilities of utilizing strong coupling effects in conjunction with large transient fields to achieve coherent control of a superconducting condensate and an avenue towards light enhanced superconductivity.