Graph decompositions for combining "local" functions of vertex labelings
What I want to say is too long for (but really should be) a comment.

If I am reading the question correctly, you want a FPRAS (fully polynomial randomised approximation scheme) for either of the above quantities, each of which includes various #P-complete problems as special cases.  In particular, you want a general FPRAS in the case of planar graphs, by the use of cluster expansion.

I doubt that this is possible due to the fact that NP-completeness of the existence problem (e.g. proper colouring) implies that the corresponding counting problem (e.g. number of proper colourings) is complete in #P with respect to AP-reducibility (approximation-preserving).  See Dyer, Goldberg, Greenhill and Jerrum, Algorithmica (2004) 38: 471-500.

But perhaps I've misread the question.

(Actually, would you be able to explain to the uninitiated the meaning of high-temperature expansions?)