For one whose custom is to drink wine (or grape juice) after reciting Birkat Ha-Mazon and to refrain from eating wine during shavua she-chal bo Tisha b'Av:
Does the requirement to use a cup trump the prohibition of not consuming wine, or should one continue to bentch al ha-kos?
By asking about Shavu'a sheChal Bo, I assume you are looking for a Sephardi perspective. Maran writes (Shulchan Aruch OC 551:10):
ומותר לשתות יין הבדלה וברכת המזון.
It is permitted to drink the wine of Havdalah and Birkat haMazon.
The Rama there notes the Ashkenazi practice of refraining from drinking both of those and instead letting a child drink it (see our discussion here).
The Kaf haChaim (a Sephardi) notes a later Sephardi practice to be stringent like the Ashkenazim only in regards to Birkat haMazon but not in regards to Havdalah.
The Mishna Berurah (sk 71) notes for Ashkenazim that in a case where no child is available, one should rely on those who do not require wine for bentching and bentch without wine (as opposed to Havdalah where he rules you should drink the wine yourself.)
