With a commercial product like oven cleaner or brass cleaner, don’t forget that you can learn a lot by reading the label. If the thurible has a manufacturer on it, you can also contact them. If you contact them, describe how the stain got onto the piece, and what it looks like in detail (color, texture, etc.), so they can better decide whether you’re dealing with a surface residue or a chemical change in the piece itself. They could be a wealth of information.
Whatever you decide to try, try it on an inconspicuous area first, to see what it does to the finish. If the finish endures the treatment without harm, then consider trying it on the area that shows. This is true for physical/abrasive measures, too. If you’re trying to physically knock off some of the excess before going to the chemicals, see how much abrasion the finish can stand without scuffing.