2SLS with a boolean regressor

So, I have the following linear model: $$y = \alpha + \beta x + u$$ and $$x \in \{0,1\}$$, i.e. the variable $$x$$ is boolean. Moreover $$x$$ may be endogenous, and I have a set of instrumental variables $$\boldsymbol{z}$$ which are exogenous. In this situation usually one uses a simple 2SLS regression and that's it. But I was wondering whether one could first regress $$x$$ on $$\boldsymbol{z}$$ thorough probit, and then take the fitted values $$\hat{x}$$ as instrumental variables in the second step of the regression, where we use $$\hat{x}$$ as instrumental variable for $$x$$ and use IV.
So I have replaced the OLS regression of the first step with a probit regression.

Is the result of this kind of two step regression consistent? Does it make sense to do so?

Thanks!