Is it permissible to pray that God makes a specific girl good for me, knowing that I am engaged and wish to marry her?

There is no objection to supplicating that Allah makes your marriage to this girl good for you, as the fundamental principle of supplication is its permissibility. The Prophet, peace and blessings be upon him, said: "O Allah, I ask You for her goodness and the goodness of what You have instilled in her." One must, however, be careful not to fall into what displeases Allah with her before the legitimate marriage contract.