Modern biological tools generate a wealth of data on metabolite and protein concentrations that can be used to help inform new strain designs. However, learning from these data to predict how a cell will respond to genetic changes, a key need for engineering, remains challenging. A promising technique for leveraging omics measurements in metabolic modeling involves the construction of kinetic descriptions of the enzymatic reactions that occur within a cell. Parameterizing these models from biological data can be computationally difficult, since methods must also quantify the uncertainty in model parameters resulting from the observed data. While the field of Bayesian inference offers a wide range of methods for efficiently estimating distributions in parameter uncertainty, such techniques are poorly suited to traditional kinetic models due to their complex rate laws and resulting nonlinear dynamics. In this paper, we employ linear-logarithmic kinetics to simplify the calculation of steady-state flux distributions and enable efficient sampling and inference methods. We demonstrate that detailed information on the posterior distribution of parameters can be obtained efficiently at a variety of problem scales, including nearly genome-scale kinetic models trained on multiomics datasets. These results allow modern Bayesian machine learning tools to be leveraged in understanding biological data and in developing new, efficient strain designs.