This is me writing out of ignorance, but I wonder how a multi-pass, thicker ink, on a textured card stock (maybe like a linen postcard style stock) would look? I think it would be awesome to duplicate the old litho process, but if it is extremely cost prohibitive then maybe something like what I described would give it the same degree of eye-appeal if not exactly the same kind.
Here is an example from the 2007 Upper Deck Masterpieces set (I incorrectly referred to it as a Topps set above). I would make them smaller and still do something refining with the printing but I think it at least conveys the potential of alternate card stocks for adding texture and depth.