We present two new results on the frequentist coverage of Bayes posteriors when estimating linear functionals of the unknown parameter in a nonlinear, PDE-based inverse problem. The results are valid for Gaussian process priors under sufficiently strong smoothness assumptions on the model. They provide a characterization of asymptotic coverage and Gaussian geometry for the standard posterior credible bands based on the first two moments, irrespective of whether Bernstein von-Mises (BvM) theorem holds or not. In the case it does hold, our sufficient conditions are sharper than previous known results. The proof is based on a particular choice of a least favorable transformation of the parameter and an injectivity estimate on the prior RKHS. As an application, we study the case of estimating diffusivity coefficient in a Darcy flow problem, where necessary conditions of BvM breaks down. This is a joint work with Katerina Papagiannouli.