I mean "with simulations using a probability distribution [for the true parameter] different from the prior used in the Bayesian analysis." (The issue of model error is a separate question.)

Yes, in this case would should conclude there is something wrong with the Bayesian way. If you hand me a statistical method to e.g. estimate some parameter that frequently returns answers that are far from the truth, that is a problem. One cannot assume the prior exactly describes reality (or there would be no point in doing inference, because the prior already gives you the truth).