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We propose a probabilistic framework for performing simultaneous estimation of source structure and fringe-fitting parameters in Very Long Baseline Interferometry (VLBI) observations. As a first step, we demonstrate this technique through the analysis of synthetic short-duration Event Horizon Telescope (EHT) observations of various geometric source models at 230 GHz, in the presence of baseline-dependent thermal noise. We perform Bayesian parameter estimation and model selection between the different source models to obtain reliable uncertainty estimates and correlations between various source and fringe-fitting related model parameters. We also compare the Bayesian posteriors with those obtained using widely-used VLBI data reduction packages such as CASA and AIPS, by fringe-fitting 200 Monte Carlo simulations of each source model with different noise realisations, to obtain distributions of the Maximum A Posteriori (MAP) estimates. We find that, in the presence of resolved asymmetric source structure and a given array geometry, the traditional practice of fringe-fitting with a point source model yields appreciable offsets in the estimated phase residuals, potentially biasing or limiting the dynamic range of the starting model used for self-calibration. Simultaneously estimating the source structure earlier in the calibration process with formal uncertainties improves the precision and accuracy of fringe-fitting and establishes the potential of the available data especially when there is little prior information. We also note the potential applications of this method to astrometry and geodesy for specific science cases and the planned improvements to the computational performance and analyses of more complex source distributions.