Is it permissible to send formal messages, devoid of any transgressions or emotional language, to a female colleague who got engaged after graduation, with the aim of reassuring her of care and commitment, within the bounds of etiquette and Islamic law, and with the knowledge and approval of her family?

We appreciate your eagerness for righteousness and avoidance of the forbidden. If you are safe from temptation and the message is sent via mail and contains nothing contrary to Islamic law, then there is no harm in sending it. However, it is preferable to avoid communicating with her by any means, lest it become a pretext for what is forbidden. Your goal can be achieved through a relative of yours (such as your mother or sister). And know that frequent supplication to Allah and piety are the best aids in facilitating matters, for whoever fears Allah, He will make a way out for him, provide for him from where he does not expect, and make his affair easy for him.