I wouldn't tumble it.
It will hurt the button, everything tumbled looks tumbled unfortunately.
You obviously like it enough to ask if you should tumble it, therefor it's valuable enough that you shouldn't. Make sense? lol
If you want to "preserve" it, enhance features, or plain just want to mess with it get a mixture of mineral oil and beeswax. This is sold as a wax for salad bowls.
You could optionally set it in some acetone first, gently pressing on it with a paint brush to remove loose soil and organics. Then apply the wax/mineral oil mixture.