This is not a direct answer to your question, just a few references. The notion of discrete curvature, and in particular, discrete mean curvature, has been studied quite a bit. Perhaps one of the best concise source is John Sullivan's 2007 paper, "Curvatures of Smooth and Discrete Surfaces" (arXiv link, PDF link). In Section 4.5, p.7, he seems to be directly addressing your situation, with the combinatorial type and the volume fixed:
A discrete surface which minimizes area among surfaces of fixed combinatorial type and fixed volume will have constant discrete mean curvature $H$ in the sense that at every vertex, $\\mathbf{H}_p = H \mathbf{A}_p$, or equivalently $\nabla_p\operatorname{Area} = -H \nabla_p \operatorname{Vol}$. In general, of course, the vectors $\mathbf{H}_p$ and $\mathbf{A}_p$ are not even parallel... [etc.]
For other literature, perhaps Konrad Polthier's 1996 paper, "An Algorithm for Discrete Constant Mean Curvature Surfaces," could give you ideas and leads. Google scholar shows that 17 later papers cite that one, including Konrad's own 100-page 2002 document, "Polyhedral surfaces of constant mean curvature" (PDF link).