You can try to relate alpha to an observed variable using the Constraint= option and then express alpha as a deterministic fcn of that in Model Constraint.
Or, you can use mixture modeling to give alpha a non-parametric distribution.
Or, perhaps this paper about non-linear growth using Mplus can be helpful:
Grimm, K., Zhang Z., Hamagami, F. & Mazzocco, M. (2013). Modeling nonlinear change via latent change and latent acceleration frameworks: Examining velocity and acceleration of growth trajectories Multivariate Behavioral Research, DOI: 10.1080/00273171.2012.755111. download paper contact first author show abstract