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The spectral lags of gamma-ray bursts (GRBs) have been viewed as the most promising probes of the possible violations of Lorentz invariance (LIV). However, these constraints usually depend on the assumption of the unknown intrinsic time lag in different energy bands and the use of a single highest-energy photon. A new approach to test the LIV effects has been proposed by directly fitting the spectral lag behavior of a GRB with a well-defined transition from positive lags to negative lags. This method simultaneously provides a reasonable formulation of the intrinsic time lag and robust lower limits on the quantum-gravity energy scales ($E_{\rm QG}$). In this work, we perform a global fitting to the spectral lag data of GRB~190114C by considering the possible LIV effects based on a Bayesian approach. We then derive limits on $E_{\rm QG}$ and the coefficients of the Standard Model Extension. The Bayes factors output in our analysis shows a very strong evidence for the spectral-lag transition in GRB~190114C. Our constraints on a variety of isotropic and anisotropic coefficients for LIV are somewhat weaker than existing bounds, but they can be viewed as comparatively robust and have the promise to complement existing LIV constraints. The observations of GRBs with higher-energy emissions and higher temporal resolutions will contribute to a better formulation of the intrinsic time lag and more rigorous LIV constraints in the dispersive photon sector.