A quick warning – Jacobians are defined for only 1-1 functions. There is not a well-defined Jacobian for a 2-1 map that takes in two parameters and returns a single parameter. Formally you have to map \theta_{1} and \theta_{2} to two parameters, say \delta = \theta_{1} - \theta_{2} and \sigma = \theta_{1} + \theta_{2}, compute the Jacobian (which here would be a constant and hence ignorable), and then marginalize out the second variable \sigma.
Regarding the question of bias – in general there is no guarantee that a Bayesian posterior, or any posterior expectation value such as a posterior mean, will be close to the true value. The figures you showed don’t seem to indicate any particularly pathological behavior.