The "mclust" package in R is a general finite mixture modeling package that can accommodate latent categorical variables (classes) and continuous manifest variables. A software review in The American Statistician (found here: http://statisticalinnovations.com/products/LGreview.AmericanStat.pdf) states that it can handle continuous variables (page 85, right column), so it seems that all the pieces are there to get an LPA to run. I haven't done it personally though ,so I don't known how difficult it may be. 

Thanks very much, Daniel. It sounds promising. I will try that and will update this post with the results.

If the heterogeneity in your sample (i.e., the latent sub group) can potentially be observed by a combination of predictors and you have no good hypothesis about their relation with the continuous outcomes, you may want to have a look at SEM Trees:

Brandmaier, A. M., von Oertzen, T., McArdle, J. J., & Lindenberger, U. (2013). Structural equation model trees. Psychological methods, 18(1), 71.

We also have an R package for running SEM Tree analyses.

Thanks, Andreas.

I have a theory for the number of clusters and hypotheses that relate the clusters to the continuous variables. I was wondering if SEM Tree and/or LPA /LCA are the best approaches for testing my theory and hypotheses. Any input is very welcome.

I've been using TidyLPA to great success

https://github.com/jrosen48/tidyLPA