We report on a spectroscopic study of the unusual Li-rich red giant (RG) recently discovered in NGC 6819. This star was observed by the Apache Point Observatory Galactic Evolution Experiment (APOGEE) survey as part of the survey's calibration cluster sample. We use the high-resolution, near-infrared APOGEE spectrum to address its questionable cluster membership and test the hypothesis that Li was regenerated by nuclear processes and mixed to the surface. Previously reported [Fe/H] and radial velocity (RV) of the Li-rich star are consistent with cluster membership, and the star's optical and infrared colors place it on the cluster's red giant branch (RGB), below the luminosity bump. Most models of internal Li regeneration on the RGB can only explain Li-rich stars at the luminosity bump, but the currently favored model for the Li-rich star is a relatively new variation on Li regeneration that can explain the star's lower RGB position. This model predicts that the ratio of 12C/13C at the stellar surface should be reduced compared to normal Li-poor RGs, a signature we sought to measure. However, the Li-rich star's recently reported asterosesmic properties are inconsistent with cluster membership. Specifically, the log g inferred from asteroseismology is significantly lower than that of similar RGs in the cluster. We find the membership question to be unresolved with our analysis — our spectroscopic measurement of surface gravity confirms the asteroseismic result, but the detailed abundances and RVs that we measure are still consistent with cluster membership. Our Li-enrichment test is more conclusive. We find a C/N ratio that demonstrates that Li dilution should have occurred, but the 12C/13C is consistent with normal dredge-up and inconsistent with Li-enrichment mechanisms that require unusually deep mixing.