Thanks for explaining about how you found that page - it was some good material but the ITensor-specific parts might be out of date now so I’ll make sure to remove it just in case.
I think your question about the behavior of MPS parameters is a good one, but one caution is: are you sure you have a good way to access the parameters in a truly gauge-independent way? You could always bring the MPS into left-canonical form, say, but there might be some left-over gauge freedom such as if some of the density matrix eigenvalues are degenerate, as can often happen when symmetries are present (whether you conserve them or not) or just from the arbitrariness of the signs of the U and V matrices in the SVD. So that’s one thing to think about.
Otherwise, yes I would expect the parameters to be continuous. One example of truly gauge-indepent “parameters” are the density matrix eigenvalues themselves, as well as the eigenvalues of the MPS transfer matrix. There have been some pretty detailed studies of how these behave at and near phase transitions (studies of the entanglement spectrum, as well as studies by Verstraete and others about behavior of the MPS transfer matrix eigenvalues). So it might be good to look up some of that literature if you haven’t already.