"You have no doubt of the mutual attachment of your brother and your friend;" that is the key phrase. What Henry is saying is that, if Isabella and James (or any couple) are truly in love, they can survive the disagreements and disruptions that affect virtually every relationship.

Look at the number of times misunderstandings have already arisen between Henry and Catherine, enough to end any casual relationship and almost to end theirs. Still, Catherine and Henry keep coming to terms and putting things back together through mutual understandings.