Bartnik data are a Riemannian 2-sphere of positive Gaussian curvature equipped with a non-negative function H to be thought of as its mean curvature in an ambient Riemannian 3-manifold. Mantoulidis and Schoen suggested a construction of asymptotically flat Riemannian 3-manifolds of non-negative scalar curvature which allows to isometrically embed given Bartnik data of vanishing mean curvature, i.e. H=0. They use their construction to explore stability of the Riemannian Penrose inequality. In collaboration with Cabrera Pacheco, McCormick, and Miao, we adapt their construction to constant mean curvature (CMC) Bartnik data, i.e. H=const.>0. Moreover, with Cabrera Pacheco and McCormick, we extend their construction to the asymptotically hyperbolic setting both for H=0 and for H=const.>0 Bartnik data. I will present the construction as well as the motivation for such a construction which is related to Bartnik’s quasi-local capacity/mass functional and its minimizing properties.