Just a comment, no solution: I think the model comparison you suggest is smart and this is a useful request.
I must disclaim that I never got into factor analysis, the confusing frequentist terminology is usually prohibitive when I try. And I also did not work on the example you link.
However, what I find intuitive, is that if you have two latent variables covering three factors, it might be that the two jump between the local optima and that is the reason for lack of convergence.
Just some thoughts:
- have you tried increasing to four latvars, to see if that works?
- is it possible to increase or decrease noise, i.e. reduce/increase sharpness of the simulated data, so that the local optima are less or more distinct? (thinking in terms of Normals here, but you get the point)