Hi Max,
Yup, that idea sounds pretty feasible with the package. You'll want to create a new LatentSemanticAnalysis object and run your corpus through it to get the dimensionally-reduced representations of your documents. Since the package doesn't normally retain the document space on its own, when you create the LatentSemanticAnalysis object, be sure to set the retainDocumentSpace argument to true. If you have new documents that you want to project into your dimensionally-reduced space, there's a project() function that will let you do that.
All of this said, LSA is old technology and you might want to try something more modern. Latent Dirichlet Analysis could work well (python's gensim or java's Mallet packages do this). Another simple approach would be grabbing some off-the-shelf word vectors (e.g., word2vec or GloVe) and then treating message as the sum of its words' vectors. That summation works fairly well for a document representation, especially for the short documents like you'll see in social media. The big word2vec dataset from google also has lots of spelling variants in it (e.g., "cool", "cooool") so it's robust to some of the variation you'll see in social media.
Hope this helps!
David