Col. Fitzwilliam appears to take an attitude to matrimony not dissimiliar to Charlotte Lucas. Of course, it will be for Lady C. to have ultimate say. He does have the 'Honourable' in front of his name. What makes me think the Colonel may consider proposing is he's already made clear he feels he cannot marry without some attention to money. It could work to advantage both ways. if Anne is so weak and sickly, she may prefer to marry (another) cousin if she feels she can trust him to care for her. Col. Fitzwilliam may feel a degree of consideration for Anne as well, apart from financial security afforded him. Anne is unused to the society of young men outside her family. Lady C. seems to like familial marriages. If she felt it to be her idea it may to give her something ese to plan for and organize down to the minutest detail!