When faced with a decision like this I generally start sketching ideas. Something "perfect" usually comes to me after a few sketches.
Paint will stay on metal, especially if you use a primer first. (and you don't smack the paint too much!)
The possibilities are endless. You probably want something without a lot of detail or contrast that doesn't take the attention away from the carved turning - I like your idea of a black base of some sort. You might consider turning a simple pedestal of some sort, perhaps not even fastened to the piece. Or make a cradle from wood, perhaps carved. Three carved or turned legs? Clear plastic might be interesting. Two simple pieces of flat wood crossed?
I once made a base to elevate a triple bowl a bit off the table. I drilled three angled holes in a simple block of wood and inserted three copper rods, angled out so the thing would sit between them. Another time I welded a single angled bar to a chunk of steel, distressed both of them, then added a thin steel rod bent to cradle the object.
Again, making some sketches would probably help a lot. If you make a base that isn't permanently attached you could mock up several ideas to see how they work, maybe raised to different heights. Experiment with supporting the piece so it sits horizontally, vertically, angled? And unless there is a pressing need, I wouldn't get in a big hurry, but let it sit and look at it and think about it for a while!
Corian is a good material to make a base for such a piece. You have a choice of different colors and you can easily glue it to the wood and turn it, or make it a separate piece that you set your wood piece on. They have solid colors or stone type materials which are an easy way to accent the display.
When you've got some sketches and/or examples from Google or Pinterest or whatever, come back with your ideas. We'd be glad to help. If you start a new thread, and start the title with "Design Critique…" or something similar, we'll know you are really looking for an open design discussion, and not just some namby-pamby glowing generalities.
We're going to try this method for encouraging productive design discussions on the Turning, Messages and Hand Tools fora. We still ask that comments be objective and polite. Please, no hurling of insults, and of course no politics or religion. In design discussions, it help us understand your comments if you identify your stylistic likes and dislikes. e.g., "I only build Chinese Chippendale" or "I hate mesquite!" Of course, I don't hate mesquite.
You start by measuring the diameter of the wire, and drilling a hole the same size in the top of the wooden base. Often, wire has a protective coating of oil on it to inhibit corrosion, or to facilitate forming it during manufacturing. You have to get it off if you want the paint to stick. To be sure you've got it all, I'd use a series of solvents: for example: naphtha, acetone, and denatured alcohol. There are other, more flammable or more toxic, solvents available if those don't work, including toluene, methyl ethyl ketone and methylene chloride, but I would not recommend them. Rustoleum in a rattle can should stick to the wire as well as anything.
What, you've never gotten an insincere, namby-pamby compliment? I know it doesn't happen on WoodCentral, but other Internet woodworking forums have their share of Charley Tunas who would ooh and aah over anything with a thick enough coat of polyurethane on it.
I think I like the platter better positioned with the grain running horizontally.