How to tumble polish manganese bronze

We have a customer who is polishing manganese bronze rings and has recently bought a tumbler to try and speed up the process. Previously, he was buffing with a reddish brown compound (I presume something like tripoli but he didn't know exactly what it was called) and then polishing with a blue compound (something like blue dialux or blue rouge but again he didn't know the name of the product). This was giving him good results but is very slow and tedious.

He has recently bought a small vibratory tumbler (no flow-through system but only small batches) and has some ceramic media (medium cut) and stainless steel shot for burnishing with burnishing compound but is not getting the same kind of results as from the buffing.

What recommendations would you have for this fellow on both media and compound. I feel that he should be able to get a finish using the media he has with an appropriate cutting and polishing compound. We would appreciate some advice on this please.