Polynomial time solvable instances of Max-Sat
This does not answer directly your Max-SAT problem but the references may guide you to the complete answer.

Szeider showed that Satisfiability is ﬁxed-parameter tractable when parameterized by the treewidth of the incidence graph. Samer and Szeider gave an eﬃcient dynamic programming algorithm. 

References

S. Szeider. On ﬁxed-parameter tractable parameterizations of SAT. In Proc. 6th
International Conference on Theory and Applications of Satisﬁability (SAT’03),
Selected and Revised Papers, vol. 2919 of LNCS, pages 188–202. Springer-Verlag,
2004.

M. Samer and S. Szeider. Algorithms for propositional model counting. In Proc.
14th Internationial Conference on Logic for Programming, Artiﬁcial Intelligence
and Reasoning (LPAR’07), vol. 4790 of LNCS, pages 484–498. Springer-Verlag,
2007.

Samer and Szeider, Fixed-parameter tractability. In A. Biere, M. Heule,
H. van Maaren, and T. Walsh, editors, Handbook of Satisﬁability, part 1,
chapter 13. IOS Press