looking more closely at your answer, could you please ellaborate a bit more on how did you arrive to the conclusion that the derivative with respect to $\alpha$ results in vega times the derivative of volatility with respect to $\alpha$? To obtain $vega$ you would have had to derive the option's price with respect to $\sigma$, so I must have missed something. Moreover, how do you conclude that $d\hat{\sigma}/d\alpha < 0$ and from there that increasing the shift shifts the vol curve down? Thanks

Thanks for your answer, very thorough and helpful. You describe how price is not affected, but what about the vega of the option? Delta and Gamma remain the same, but intuitively I guess that is not the case for vega?