Is it permissible to propose to a girl who has agreed to marry another suitor, with whom a formal engagement has not yet taken place?

There are no formal rituals required for the engagement to be established; rather, the agreement of both parties and their inclination towards each other suffice. If this is achieved, then it becomes forbidden to propose to a woman already engaged to another, unless the suitor abandons the engagement or gives permission. The girl and her guardian are best aware of what transpired between her and the prospective young man. If her engagement becomes difficult, then seek another, and seek Allah's help to grant you a righteous wife.