The analogue of 2-tailed 95% CI would involve the fraction of fits where either the 2.5% bound is greater than zero or the 97.5% bound is less than zero.
My guess is that your priors are responsible for the bump in apparent power. What happens if you switch to a Gaussian response (equivalent to putting all the prior mass on the degrees of freedom at very large values), with a very weakly regularizing prior on the standard deviation (say, normal(0, 5))?