I started out by applying shellac to the binding and the white rosette with a Q-tip. This 'masked' these areas from the dye. Then I went around the rosette with the red dye mixture.
After that, I applied red to the whole top, then brown.
As the water in the dye starts to evaporate, the wood loses some of its intense coloring, but that will come back with the finish.
When the water is just about dried, the top matches the back and sides very well.
Here's what the newly dyed top looks like compared to the sides. This also gives an idea of what the binding will look like.