You have a couple of options.
1. Send the putter to a restoration company, there are several to choose from.
Google - refinishing Bettinardi putters.
They would certainly offer a guarantee on their service and it would look like new again.
Probably a pricey option.
Check out - dreamputters.com www.golfwrx.com/forums/topic/4332-bettinardi-refinish
2. There are methods of restoring small blemishes on metals.
You can find a lot of information search " gun bluing"
An inexpensive procedure would be to first try applying a touch up finish such as either of the following products.
The Birchwood Casey Presto Gun Blue Pen is a fast and easy way to apply a blue black finish to your carbon steel firearm. The non-drip, no mess felt tip applicator quickly fixes bluing nicks and wear. Will not work on stainless, aluminum or other non-ferrous metals.
The Birchwood Casey Perma Blue Finish is a room temperature chemical that is used to touch-up worn areas on carbon steel. The final color is a blue - black and will darken with additional applications. As with all chemicals, care should be taken and protective gloves and eye wear should be used. 3 oz.
If the chatter has deep impressions, you could try using a very fine mill file to remove the raised dings.
Finish off with a scouring pad using it lightly on the damaged surface.
I would suggest practicing first on an old forged club, to see your results.