Is it permissible to adopt the weaker opinion regarding the suckling of an adult to establish mahramiyyah between a man and a woman in this specific case, given the man's desire to become a mahram to the woman he cares for who is facing a serious illness?

The preponderant opinion is that adult suckling does not establish a prohibitive relationship (mahramiyyah), and this is the view of the majority of scholars. Your heart's attachment to her or your need for her does not necessitate adopting this view. Rather, you must keep away from her to avoid temptation. If you wish to help her, you can marry one of her daughters, which would make you a mahram to her, or you can seek out husbands for them. We do not see it permissible for you to suckle from this woman or to have close contact with her.