We present a precision analysis of the Xe136 two-neutrino ββ electron spectrum above 0.8 MeV, based on high-statistics data obtained with the KamLAND-Zen experiment. An improved formalism for the two-neutrino ββ rate allows us to measure the ratio of the leading and subleading 2νββ nuclear matrix elements (NMEs), ζ312ν=-0.26-0.25+0.31. Theoretical predictions from the nuclear shell model and the majority of the quasiparticle random-phase approximation (QRPA) calculations are consistent with the experimental limit. However, part of the ζ312ν range allowed by the QRPA is excluded by the present measurement at the 90% confidence level. Our analysis reveals that predicted ζ312ν values are sensitive to the quenching of NMEs and the competing contributions from low- and high-energy states in the intermediate nucleus. Because these aspects are also at play in neutrinoless ββ decay, ζ312ν provides new insights toward reliable neutrinoless ββ NMEs.