I recently demeaned my dependent variable by a group interaction. Mainly because simply adding them as an interaction was detrimental for the calculation time of my regression and caused singularities when estimating a Tobit Regression.
The added benefit was that it lead to a much smoother distribution of the dependent variable.

I later read, in this paper by Gormley and Matsa (2014), in the Review of Financial Studies, that this usually leads to an attenuation bias (as opposed to including the interaction normally).

I have going to this blog, with some enlightening answers by Mr. Wooldridge, and I was wondering if this attenuation bias can somehow be fixed by a control function approach (also called 2SRI I believe),
or whether that if even necessary if I am using an IV anyway.

I want to use the control function approach, because I might want to use a distribution for which Stata does not have a suitable 2SLS.
If I understand the posts correctly, the control function approach, can also be applied when the second stage is non-linear.

Quote by Mr. Wooldridge

If you use the control function approach then you need to estimate a reduced form for each unique EEV. Putting in nonlinear functions of those EEVs requires no change: you just add the first-stage residuals to the second stage GEE. Now, you might want to include more functions of the residuals, such as squares and cross products. This makes the CF approach more flexible.
As a small remaining question, what is the downside of the control function approach?