How can you clean bronze?

Yes with Bronx cleaner

You can use a l pad with soap in it like Brillo or SOS to clean the bronze. Then wax the headstone with a clear paste wax. That will help the water bead up and roll off, rather than drying on the metal.

Get bronze cleaner

Try plain white toothpaste and a soft cloth

You can make a paste of baking soda and lemon juice. Then make sure you wax it with paste wax to keep it form spotting.

Did you try Magic Eraser sponges? I used it to clean up a very old brass piece, it was literally black, it did wonders!

I would ask the cemetery people and the marker provider. This is their profession and business.