It's really not that difficult to do yourself, starting with higher grits of wet and dry paper, and working your way down until you get the finish you like. I used to polish the slides of pistols for friends a long time ago, leaving any recessed features alone. You need to be prepared for it taking a long time though. I've polished various items before in my workshop at work where it was started by the owner and abandoned because they didn't have the patience!
Do you have any Engineering workshops around you? There are tons in my city and they often charge reasonable prices. I've never done a Provari, but the hard part would be removing the screen and board. I would be tempted then to put it in a lathe and simply push sheets of wet and dry against it, and finish with a soft buffing wheel and a touch of jewellers rouge.