I guess if you could vectorize the gradient, even without worrying about the dot texture of the original, just doing a smooth gradient, then if you had it printed using the silk-screen or offset printing process, they could use the halftone process for the gradients.
I assume that glittery silver is just a type of printer's ink that any print shop using traditional methods would have or have access to?
So what it looks like to me; and I only have a vague understanding of the silk screen process, and pretty much no understanding of the offset printing process (haven't really read up on it yet); is that one pass would lay down the black background, another pass would lay down the red that surrounds the letters, another would lay down the black outline of the letters, another for the glittery silver areas, then the white center of the letters (unless the material was already white, in which case that step wouldn't be necessary) and finally the halftone gradients. Does that sound about right?
Duplicating the halftone gradients in a raster file in Photoshop would be useful only if you were going to do an inkjet printout I'd imagine, which would be okay at best, and certainly missing the glittery silver of the original. Plus, if you were going with raster for an inkjet printout, you might as well just use a cleaned up high-res scan of the original in the first place, which would already have the halftone gradients in it.