The first and easiest way is to find a flat metal washer about the same size as your missing domed washer and color it with a metallic gold sharpie. It usually takes me 3 coats of the sharpie, letting it dry an hour or so between coats to get good coverage. What do you think?Placed in the shadows under the horse, I do not think the replacement domed washer is very noticeable.

The second method is a bit more involved, but the results are much more convincing.  I start with a washer just as in method 1, but then I add plastic wood which you can find at the hardware store (I prefer DAP brand which is similar to clay or putty and water clean-up.)

I encased the washer in a blob of the plastic wood. You want to make your plastic wood blob quite a bit bigger than the dome washer you are trying to replicate because you'll be sanding & shaping it after it dries fully.  The color of the plastic wood will go from beige to white (there is also a color changing plastic wood that is pink when wet, but will also dry white.)  However, in this particular step you can't go off the color of the outside, you need to give it enough time to dry fully all the way through the blob - I would not wait less than a day or two, I prefer to wait 2 to 3 days because it is pretty thick.

The next step is to drill a hole in the center of your plastic wood blob. You'll want to find a drill bit that fits comfortably inside the hole of your washer and be sure to drill a small starter hole then work your way up to the largest size that fits through the washer so your plastic wood blob doesn't shatter.

Once the hole was drilled into the center of the washer, I recommend to test that the inner post/pole (the part that goes through the horse's body) fits through the hole you've drilled. I then sanded the bottom flat to the washer inside the blob. This will give you a solid base that will sit flat against the horse's body.

Next I sanded the width of the blob until it was close to the size of the washer. I left a bit of extra so the washer (hopefully) doesn't show through after I sand the blob into a dome shape.

I also quickly sanded the top of the plastic wood blob down flat until it was just a bit taller than the original dome washer. The flattening parts can be done quickly by laying a piece of sand paper on a flat surface like your countertop, then just rub the top and bottom of the washer blob back and forth across the sand paper until you've gotten them flat and to the right height.

The next step will have to be done by hand, that is shaping the blob into a dome shape that matches your original dome washer.

This part takes a little bit of time, I recommend going slow and comparing the shape of your replacement from both the top and sides to the original brass dome washer until you have the right shape.

Once you have your replacement dome washer shape done, you can use a little bit of water to smooth out the plastic wood. You'll notice the plastic wood change to a tan color when it's wet, you'll want to wait until it fully dries back to the white color before the next step. I left mine to dry for a couple days because I had other things to do, but I think overnight is probably long enough.

Once your replacement washer has dried fully, it's time for painting. You could use whatever paint you want, metallic spray paint or a metallic acrylic craft paint in an antique brass should work. I prefer to use my metallic gold sharpie. I did 3 or 4 coats, letting it dry an hour or so between coats.

Once your paint is dry you can reassemble your carousel figurine. I like to put the replacement washer on the bottom to kind of hide any color-match deficiencies in the shadows under the horse.

I believe one would be hard pressed to notice that this dome washer is not the same as the original without being told it was.