Multifractal analysis is a reference tool for the analysis of data based on local regularity and has proven useful in an increasing number of applications involving univariate data (scalar valued time series or single channel images). Recently the theoretical ground for a multivariate multifractal analysis has been explored, showing its potential for capturing and quantifying transient higher-order dependence beyond correlation among collections of data. Yet, the accurate estimation of the parameters associated with these multivariate multifractal models is challenging. Building on these first formulations of multivariate multifractal analysis, the present work proposes a Bayesian model and studies an estimation framework for the parameters of a quadratic model for the joint multifractal spectrum of bivariate time series. The approach relies on a novel joint Gaussian model for the logarithm of wavelet leaders and leverages on a Whittle approximation and data augmentation for the matrix-valued parameters of interest. Monte Carlo simulations demonstrate the benefits of the method with respect to previous formulations. In particular, we obtain significant performance improvements at only moderately larger computational cost, for large ranges of sample size and multifractal parameter values.