I tried a CBB a few years ago and was never able to get it eating. It would pick at the rocks non stop but at least for me that was not enough to sustain it. I tried all of the frozen foods that people recommended and mine never showed any interest in it. After a few weeks it finally found something in the tank that it would eat, hammer corals. I moved it to a qt tank and it died shortly after.
A year ago I got another CBB but this time I also had a culture of live white worms. I see several others in the thread have already mentioned live white worms and in my experience they are the best food to entice a copperband to eat. Mine was kept in qt for a few months until I could get him eating worms out of the water column, instead of only eating them once they landed on the rocks and bottom of the tank. Once I moved mine to my display tank he was chased by several of my fish (including a few normally docile fish). I used the mirror trick which at least kept the purple tang occupied. My CBB was confined to a corner but every day the aggression got slightly better. I would initially feed the fish their normal food on one side of the tank and then squirt some live white worms in front of the CBB on the other side so that he could eat. Eventually the aggression completely ended and the CBB is just as aggressive at getting food as any of my other fish. A year later and he will eat some of my diy frozen if he has to, but I still feed live white worms daily that I can see him looking for when I feed. Also I have/had a lot of aiptasia in my 125g display, and for a few weeks he wouldn’t eat them. Once he finally started eating them the tank went from over run to no aiptasia within a week.
All of that long winded reply was essentially to say don’t assume that picking at the rocks alone is enough to sustain them, it may be but it isn’t always. If/once he starts eating aiptasia it will probably be a short lived food supply. If you have a source to get white worms you should try them. Unfortunately mostly what you buy online is just a few for to start a culture and they usually don’t fair to well shipping during the hot part of the year.
Good luck I hope you get yours eating. They are an amazing fish and definitely worth the extra effort.