Short answer : No, you can't. That is indeed forbidden regression what you're trying to do.
Possible solutions : some brief discussion, google the keywords to learn more.
- Use LPM(linear probability model) - just use regular 2SLS with your data. sounds crazy, but a lot of people do this in economics literature. Make sure that you get the standard errors right.
- MLE - specify the distribution of your first stage equation, jointly with the second stage binary outcome equation. (i.e. use probit model with bi-variate normal errors)
- Control function approach - basically add the residuals from the first stage equation to covariates in the second stage.
for details, see Blundell & Powell(2004) or Rivers & Vuong(1988).