It is said that the best cheese for khachapuri is Imeretian chkhinti-khveli — fresh, young, not too salty. The closest substitution for it in the U.S. is fresh mozzarella curds. Often even in Georgia, a little saltier and more mature local cheeses are used for convenience. Asadero and Feta are the easiest to find where I live. Greek Manouri cheese is my favorite choice recently. Experiment with the types of cheese you have available and adjust the filling to your taste.