How dirty is the water he's in currently?
When acclimating with a cup, you'll want to float the cup in the tank you are moving him to. Float it for awhile if the temp in the new tank is different than the temp in the old one so his water in his cup will adjust. Periodically add a small amount of water to the cup. If say for the first hour floating him, add a spoonful of new tank water to the cup every fifteen minutes. For the second hour, add every ten minutes, and for the third hour or half hour, add every five minutes. You could slowly increase the size of the spoonfuls each hour as well. Check the temp in the cup with a thermometer and the temp in the tank to be sure its the same.
Some bettas will get stressed from being cupped. Mine got stress stripes when I cupped him and moved him for treatment. If the water in his current tank is dirty and hasn't been changed in a while, the pH might be lower than the pH in the new tank and he will need to be adjusted slowly. Moving him quickly back and forth between tanks will stress him more.
Sometimes a little stress is unavoidable. After you finish acclimation and release him into the new tank, give him some space and darkness for awhile so he can adjust. Instead of dumping him into the new tank, just hold the cup sideways into the tank and eventually he should get curious and swim out of it on his own. I would wait for him to regain color before trying to acclimate him again.