340 grams sounds perfect for me if I don't have to mess with lead tape. My target weight is 11.9 to 12oz. I'll probably string it up with Babolat Xcel 17 and if that's underpowered, then I'd use X-One Biphase 17 instead. String dampener wise, I use whatever the racket comes with and in this case it would be the little W dampener.
I'm flexible with the leather grip type. I'm avoiding thicker leather as that rounds off the bevels too much, so thin seems to be the choice. Of all the thin leather grips, which one is the heaviest? Then I'd wrap the leather with Wilson's white overgrip because it's reasonably tacky and not too thin. I like some cushioning over the leather.
I guess little dampener, just a ribbon tie or no dampener is what this racquet will ever need. But you will see whether you will lack some mass in the head, it's not just the weight, it's the distribution/balance too.
What racquet do you use otherwise and what are the specs?
I usually use the lightest basic grip around, then play with distributing the mass (lead tape, putty) at two spots, at the top of the handle and at the butt. It can get the same job done with least mass added. With heavy grip you sometimes get to situation when the effect is not desireable while with the light grip you can play with quantities and get what you want. Already present mass distribution dictates what you can do. I guess the easiest is to do what
@vsbabolat does, to reproduce exact specs of your preferrence with which you play well (weight, SW, balance), because they should also define how racquet swings, so getting all three specs to a unit should give you a familiar racquet.
