Is the Salāt al-Istikhārah the only solution, and what are the appropriate supplications for a young man to marry a girl whose family's customs prevent this marriage, even though she is of good character and religious?

Do not tire yourself, for the matter is in the hand of Allah. So, supplicate often, especially with the Istikhara prayer. If the woman is destined for you, she will not escape you. Therefore, submit to divine decree and be content with what Allah has apportioned for you.