I wouldn't bother with a carbon bar if stiffness is your aim. You won't find a carbon bar to fit that type of stem. You'll need to buy an a-head converter and an a-head stem, plus the carbon bar itself. Then you'll find that despite having a really stiff bar, the front end will still flex - the steerer, fork blades and headtube can all flex too. And so can the front wheel. So if you want to have a really stiff front end, you'll need a different bike!
I would just enjoy it as it is. Looks like a nice bike.