Dear Anders Kock - Thanks very much for your suggestions! Giuseppe Rosolini also recommended Dubuc's work, so definitely I'll look at that, as well as the other references you mention. Rosolini also recommended the book by Wyler.
Note that the "differentiable spaces" of Chen (and the general machinery of Dubuc) deal with a COVARIANT determination of structure (i.e. the structure is given in terms of certain plots INTO a set/space), whereas the one considered by Mostow is CONTRAVARIANT (structure given in terms of certain functions OUT OF the set/space).
Right. Chen's approach seemed a bit nicer for internal homs, since products are easily defined by saying a plot R^n -> A x B is a plot in A and a plot in B, and then we can define the internal hom by saying a plot R^n -> hom(X,Y) is a smooth map R^n x X -> Y. Perhaps this is an illusion of some sort, and the contravariant approach is just as good? One might fear that the covariant approach, being nice for products, would be bad for coproducts. But, one can take advantage of the fact that R^n is connected to say a plot R^n -> A + B is either a plot in A or a plot in B. Best, jb