Is it permissible to inform an engaged girl—through an intermediary—of the desire to propose to her if she breaks off her first engagement? Is this considered proposing over another's proposal? And what is the solution for informing her of this desire?

You are not permitted to express to a betrothed girl your desire to propose to her, because Islamic law prohibits a man from proposing when another man's proposal has already been accepted and both prospective spouses are content with each other. A man's conversation with a woman who is not a mahram (definition) to him is permissible only out of necessity and should be limited to that necessity. One must be wary of the steps of Satan, which can lead to sin. Furthermore, exchanging gifts between non-mahrams is not permissible if it is intended to solidify an unlawful relationship, as this constitutes consuming wealth unlawfully.