Recommend going back to the still. With the vodka, run it again, this is easy, all 24 plates. If the intention for your rum is white, you can redistill as well, but only run through your short column (4). In both cases, your focus is going to be on the tail cut. This is entirely based on your comment of visible clouding, which you should not be seeing. This is making me think that carbon will *not* be efficient here, as you'll quickly overwhelm the adsorption capacity, and waste a lot of carbon to get where you are wanting to go.
Chill filtration, there's not a whole lot out there for the craft market, most I've seen have cobbled together their own systems out of jacket tanks, freezer chests, plate and frame filters, etc. As far as something turnkey you can just ring up and order? I've never seen one. I'd love to see something work with a smaller 10" Code-7 style filter housing, as opposed to trying to run a smaller volume of spirit through a gigantic plate and frame, losing 25% of my spirit volume in the process.