Given a metric g on the 2-sphere S2 with Gaussian curvature bound below by −3, and non-negative constant H, we construct asymptotically hyperbolic manifolds whose boundary is isometric to (S2,g) and has mean curvature H (with respect to the inward-pointing unit normal). These AH manifolds have mass that is controlled in terms of g and H, reducing to the hyperbolic Hawking mass of (S2,g,H) as g becomes round or H tends to zero. This gives an upper bound for an asymptotically hyperbolic analogue of the Bartnik mass. The construction is based on work of Mantoulidis and Schoen, where they used similar ideas to effectively compute the (usual AF) Bartnik mass of apparent horizons.