Is what the mother did—in disclosing her daughter's desire to marry her cousin to her aunt—considered a mistake and an insult to the daughter's dignity, and is there a supplication for facilitating marriage?

A mother offering her daughter in marriage to a suitor is a legitimate act and not an affront to dignity. It is permissible for a woman to offer herself to a righteous man, with certain regulations. Do not grieve over missed marriage opportunities, for perhaps Allah will compensate you with something better. Maintain good assumptions about Him. There is no specific supplication for facilitating marriage, but supplication in general is beneficial.