For the extremely bright lensed galaxy SDSS J1723+3411 at z = 1.3293, we analyze spatially integrated MMT, Keck, and Hubble Space Telescope spectra that fully cover the rest-frame wavelength range of 1400-7200 A. We also analyze near-IR spectra from Gemini that cover H alpha for a portion of the lensed arc. We report fluxes for 42 detected emission lines, and upper limits for an additional 22. This galaxy has extreme emission-line ratios and high equivalent widths that are characteristic of extreme emission-line galaxies. We compute strong emission-line diagnostics from both the rest-frame optical and rest-frame ultraviolet (UV), to constrain physical conditions and test the spectral diagnostics themselves. We tightly determine the nebular physical conditions using the most reliable diagnostics, and then compare to results from other diagnostics. We find disappointing performance from the UV-only diagnostics: they either are unable to measure the metallicity or dramatically underestimate it; they overestimate the pressure; and the UV diagnostic of ionization parameter has a strong metallicity dependence in this regime. Based on these results, we suggest that upcoming James Webb Space Telescope (JWST) spectroscopic surveys of galaxies in the reionization epoch should invest the additional integration time to capture the optical [O ii] and [O iii] emission lines, and not rely solely on the rest-frame UV emission lines. We make available the spectra; they represent one of the highest-quality emission-line spectral atlases of star-forming galaxies available beyond the local universe, and will aid the planning of observations with JWST.