that follows from the dependence of U on A and B being smooth enough (across the kind of change that dM is capable of making) and nothing else.

This is incorrect. Being smooth is not a license to freely replace function values with linear approximation values; that's not what smoothness means. You have to analyze the error term, most easily presented as the higher-order remainder in the Taylor approximation. Such an analysis is what I'd tried to supply in the post I linked to.