2SLS with endogenous interaction terms
Let $X$ be the original regressor matrix and $Z = [z \;\; zd]$ be the intstruments matrix, and consider the IV estimator

$$\hat \beta_{IV} = (Z'X)^{-1}Z'y = (Z'X)^{-1}Z'(X\beta + u) = \beta + (Z'X)^{-1}Z'u$$

As long as the things that are included in $Z$ are orthogonal to the error term, we get consistency... provided that $Z'X$ remains invertible, something that you can easily check in your case.