The question asks about coating chrome paint, but you really need to take a step back to the chrome finish, itself. Applying a chrome-looking finish to plastic has always been a challenge. Before you get to trying to top coat it for protection, it's tough to just get a mirror finish. If you are able to apply a mirror finish, then you face additional challenges of keeping it that way. Some finishes will darken. Some will eventually develop matte areas. Some will wear away if they get any handling. Some will degrade if exposed to sunlight. The need for protection, the type of protection required, and what protective coating is compatible will depend on what you used for the finish.
One approach is to create the finish using something that doesn't require a protective coat. A few possible ways to do this:
Paint that looks like polished chrome is made using a silver-colored metal dust of extremely small particle size. If you mix some of that metal powder with a two-part epoxy top coat (designed for exterior use), you may get a polished chrome-like finish that is tough and stable enough to not require additional coating. This approach is something you would need to research further for precise implementation details.
There is chrome vinyl wrap that is commonly used on car bodies. It will conform to the complex curves of a car body, but I'm not sure how practical it would be to apply it to a small emblem with a lot of tiny 3D details and 90° edges on what I assume is a round shape. Warming the film and vacuum forming it over the emblem might work. But again, this would take some research and trial. This video, How to vinyl wrap with chrome, will give you an idea of how the wrap behaves in normal application on a car body.
Warming it and vacuum forming it over the emblem would provide more conformity, but it would need to be tried to see how well it works. Small, inexpensive vacuum formers are available or can be made. For this application, you just need a vacuum table (a box with a grid of holes that would probably work with a shop vac); the controlled heating needed to almost melt thick sheets of plastic would be too much for this kind of film.
If you start with chrome paint, Elmy's answer is pretty much the bottom line. This video, Protecting Alclad Chrome - 14 Clear Coats Tested, compared 14 kinds of clear coats on a specific chrome paint. The results are probably applicable to other paints. Acrylic floor polish did one of the best jobs and was inexpensive compared to most of the other options. But the testing did identify a few other products that were very good or better. One was a two part epoxy, one was a specific polyurethane top coat, and the others were clear coats designed for this purpose. Note that the coatings were at least superficially tested for durability by trying to scratch them with a fingernail, but they weren't tested for how the materials hold up to sunlight.
BTW, in almost all cases, real durability requires allowing plenty of time for the paint to fully cure before coating it (on the order of a week, sometimes more), and allowing even more time for the clear coat to cure.
Also, in case you weren't aware of it, most experienced modellers don't spray chrome paint directly on the plastic part. They start with a base coat of a gloss black paint that bonds with the plastic and dries to a very smooth surface. The chrome paint is applied to that after it's cured (typically more than a day). It doesn't take a thick covering of chrome paint over the black to produce a chrome look.
But the main idea I wanted to cover is something unusual that might be the best solution, that I stumbled across in this video, Chrome Finish on Plastic- Did I Find the Answer?. Coincidentally, the application in the video was chrome finishing a plastic hood ornament for a car. The video covers a number of products that were tested, but the surprising one was something sold as a system for fingernail polish.
You apply a clear, UV-cure acrylic resin that self-levels to a very smooth surface, and you harden it. Then you brush on a very fine colored powder (silver was one of the colors in the kit). A thin layer of the powder bonds to the coating and looks like polished metal.
The acrylic coating is tough (and immune to sunlight UV). I don't know how much handling the bonded powder would take before it wears off. But if it needs protection, my assumption is that this finish would be much easier to protect than chrome paint, with a bigger range of options. In the video, the author comments that the finish isn't fragile like chrome paint and thought there was no need to add a protective top coat on the hood ornament.
One other possibility is chrome plating the plastic. I thought this wouldn't be practical at home, but a Russian guy has a video, Few people know this secret of plastic chrome plating!, of how to do it with readily available materials that don't involve serious hazards. I'm not up on my chemistry enough to know what metal actually gets plated onto the plastic. In his process, the final plating step uses a solution made from stainless steel, and a lead electrode. I don't know if lead is used so it doesn't bond with all the ions in the solution, and the part gets plated with stainless steel, or the solution mainly serves as an electrolyte, and the part gets plated with lead. But it turns silvery and that may be all that matters. :-)
The author claims the process pulls chrome from the stainless steel into solution, and that's what gets plated. In another video, he plates an aluminum part with essentially the same process, but uses a stainless steel spoon instead of lead as the electrode. So the metal plated onto the part probably isn't lead.
