The divide between tractable and intractable was shown by Sorin Istrail to be between planar and non-planar graphs. Since 2D surfaces can have non-planar graphs imbedded in them I think (correct me if I'm wrong) planar RBIMs are what you want to look at.

That said, Istrail showed that for {-J,0,J} couplings the ground state as well as the partition function can be efficiently calculated. I'm not sure of the exact details, but the calculation of the partition function can be mapped to the calculation of the determinant which is efficient. The specifics are available as references in the paper below.

Istrail's Paper: http://www.cs.brown.edu/people/sorin/pdfs/Ising-paper.pdf

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