Is it permissible to have multiple intentions when fasting the first nine days of Dhul-Hijjah—fasting a vow and fasting them for their فضْل (virtue)—while breaking the fast on the tenth day and completing the fast for the rest of the month? And by doing so, would the vow be considered fulfilled?

It is permissible to fast the month of Dhul-Hijjah, breaking the fast on the days of Eid and the days of Tashreeq, then completing the four days missed from the month of Muharram. When fasting the nine days of Dhul-Hijjah and the Day of Arafah, one can intend to fast a vow while hoping to obtain the reward for both simultaneously.