You could actually do this using CBC/HB or logit (you would run logit in our latent class module as a model with one class).
The latent class model with a single class is a general purpose logit package. You will need to export your data and add effects or dummy-coded demographic variables, which will vary across but not within individual respondents. Then just run the model and interact the demographics with your design variables.
With HB it might be easier to be more exploratory. First, run your HB analysis. Now, using the zero-centered diffs utilities as your dependent variables, run analyses of variance to see whether those utilities differ significantly by demographics.