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Counting the number of bipartite matchings is #P-hard. Thus every #SAT problem can be reduced to counting the number of bipartite matchings. If a SAT problem is unsatisfiable however, it will have 0 solutions and thus the corresponding bipartite matching problem will have 0 solutions which is detectable in polynomial time (since a single bipartite matching can be found in polynomial time).

This would enable us to solve SAT in poly-time. What's going wrong with this reasoning?

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    It's a different notion of reduction. It is not true that the reduction maps unsatisfable SAT instances to 01 matrices with permanent 0. See http://cstheory.stackexchange.com/questions/8749/permanent-is-p-complete2011-11-09

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