Keep the nature of the dependent variable in mind for this. You haven't specified this, but " P(Y|x, A) =!= P(Y|x, B)" seems to suggest binary. If it's a variable that will call for a logitform of regression (i.e. binary, ordinal, categorical), it's not as simple as just testing equality of coefficients as the differences in coefficients may be manifestations of unobserved heterogeneity (see allison 1999, williams 2009). Personally I think you're splitting hairs regarding the interaction. Interactions aren't just significant when the signs are different, they're significant when there is a significant difference between them. That typically is the case when signs are different, but it does also occur when the signs are the same. Plus, with the nature of interaction interpretation, you can't necessarily go by the signs of the main effects themselves, they all have to be interpreted together. As a result, sometimes even when signs look opposite, they're really not because of the way the variables have to be considered together.

Is the n for each group large enough to run separate regression models for A and B and then use Clogg Tests to examine inequality in the coefficients?