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In practice (at least in the rates world), $\beta$ is preset and $\alpha$ is solved for to calibrate to the atm vols $\sigma_{ATM}$ (which are the most liquid and reliable of the market data available). For instance, in the case for normal vols and assuming a normal distribution of the fwds, $\beta=0$ then $$σ_{N,ATM}=α\left(1+\frac{2−3ρ^2}{24}ν^2T\right).$$ The $\rho,\nu$ parameters are then obtained via the sort of optimization routines you describe in order to incorporate the skew.