Interacting covariates with the instrument in the first stage
Short answer: No.
Your model is $Y=\alpha + \beta X + \varepsilon$. Even when $X$ is exogenous, if you regress $Y$ on $X$, $W_1$ and $W_2$, then the OLS estimator is inconsistent (for $\beta$) unless $W_1$ and $W_2$ do not affect $Y$ on average (after controlling for $X$) or $X$ is uncorrelated with $W_1$ and $W_2$. When $X$ is endogenous, there is no reason why your strategy should work.
Now suppose that your model is $Y=\alpha + \delta X + \gamma_1 W_1 + \gamma_2 W_2 + \varepsilon$, instead. (This is not your model; $\delta \ne \beta$.) You are saying $X$, $W_1$ and $W_2$ are endogenous, while $Z$ is exogenous. You need instruments for $X$, $W_1$ and $W_2$, but you don't. The parameters are not identified.