Is it possible to re-correspond with a girl who refused marriage for fear of repeating a failed experience? Is it permissible to pray that she becomes the questioner's share, and does that diminish his standing?

What was mentioned about the girl's good character is fine, but it is necessary to ascertain her religiosity, her adherence to obligatory duties, her avoidance of forbidden acts, and her commitment to modesty, based on the Prophet's (peace be upon him) saying: "A woman is married for four reasons: for her wealth, for her lineage, for her beauty, and for her religion. So, choose the religious woman, and you will prosper."

There is no harm in proposing to her again after inquiring about her from trustworthy individuals. If they praise her, then propose to her.

The appropriate supplication is the Istikhara prayer (prayer for guidance), to entrust the matter to Allah Almighty, for He knows the consequences of affairs. "And perhaps you dislike a thing and it is good for you; and perhaps you love a thing and it is bad for you. And Allah knows, while you know not."

If Allah does not facilitate marriage to her, then turn to Him and ask Him for a righteous wife, and seek assistance from trustworthy individuals in your search.