It would be possible to print the already halftoned image as it is without running it through another halftone process. Much like how text is not halftoned, but can include very small accurate shapes. I’m sure this makes the preproduction more complicated, but it’s essentially a software problem, not a hardware problem.
If you can extract the original halftone dots from the image, sure, you can create new mask layers using those.
But it's unlikely that you can do that, as, unless you're printing in a four-layer pure-CMYK process (and why would you?), even the best photo scanners will be unlikely to be able to capture enough information to tease apart the original offset-layers in their original colors, where halftone dots have overlapped to produce new mixed colors.
Also, you'll be unlikely to reverse the color-bleed effect of the paper to get out the original halftone dot pitch, in order to print new dots that bleed out to the same size the original's did on new, different paper with new, different inks.
Basically, this is as hard as any art forgery project (i.e. needing to match historical materials and processes and environmental conditions), but before you can even start the forging process, you need to destroy even more information by passing the image through the grid-sampling process of a scanner, and only then try to reconstruct layered vector circles from the resulting sample grid.
