The API should have a function to return the page content with all templates expanded, similarly to templates=expand in index.php, see http://www.mediawiki.org/wiki/Manual:Parameters_to_index.php#Raw.
I have already created a patch for this, however I'm unsure about the parameters that should invoke this function. It is an extra property to Revisions. It could be implemented as rvprop=expanded_content. Or there could be another rvexpandtemplates parameters.
I have attached a patch for the latter one.