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This article aims at the acceleration of an inverse heat transfer problem solution within the Bayesian framework. The physical problem involves a spatially varying heat flux, which can reach very large magnitudes in small regions, such as in the heating imposed by high-power lasers. The inverse problem of estimating the imposed heat flux is solved by using the Markov chain Monte Carlo method, with simulated transient temperature measurements. The solution of the inverse problem is based on a reduced model, which consists of an improved lumped formulation of a linearized version of the original nonlinear problem. Two different priors are considered for the sought heat flux, including a total variation density and a Gaussian density. The Gaussian prior is based on the physics of the heat conduction problem. Parameters appearing in both priors are also estimated as part of the inference problem in hyperprior models. The Delayed Acceptance Metropolis-Hastings (DAMH) Algorithm and the Enhanced Approximation Error Model (AEM) are applied with the objective to improve the accuracy of the inverse problem solution. [ABSTRACT FROM AUTHOR]