Recalculate residual before estimating RMSE

[sophiamm22]

I'm trying to do an indirect 2SLS IV estimation by running first stage regression and second stage regression separately, allowing for different sets of controls. I need to adjust the standard errors estimated in the second stage in this case. Is there a way to manually calculating the residuals, and then use reghdfe to estimate the rest, e.g. C.I., p-value, standard errors, taking into account the fixed effects and clustering?

Here's an example without fixed effects and clustering SE. I don't know how to replicate this method with FEs and Clusters, however. Thank you so much for help or any insights!