We must be patience with our newly acquired critter. I understand the concern of a newly added fish not eating but given the parameters are good and there’s nothing harassing it, it will eat in time.
With fish that are reluctant to eat like some angelfish, it’s best to keep the fish in a self contained box like an acclimation box. Once in the box, you can monitor it more closely and target feed it without the food being eaten by something else or rot away.
I use the 400C to acclimate new fish until it eats food out of the water column aggressively.
https://yourfishstore.com/collections/fish-traps-acclimation-boxes
You can also get the fish out to it’s own tank but is hooked up to the system like a refugium. But I like to use the acclimation box because it gives the fish a chance to “see” its new environment and the existing fish a chance to get used to the new comer.
Angelfish can be tricky to get to feed. using live steamer clams, and oyster on half shell can entice it to eat. Some just like flakes. Keep trying the steamer clams in the box. My theory is that all fish are curious animal. When they’re in a box, they need something to do. With nothing else to peck, the angelfish will eventually peck on the clam/oyster on half shell and get the taste of it. Eventually it will finish the clam. Now it’s your turn to switch up on the type of food. Once I see it has feed on the clam, I’ll try the food that I feed to the tank. And once it eats that pretty aggressively, I’ll let it go free. This method is not 100%. I’ve lost fish that are eating in the box but then never come out of the rock. But my success rate with this method is a lot higher than just throwing the fish in there wihout knowing that it will eat.
Flame angels should be pretty easy to get to eat. I’ve had regal angelfish that went for 3 weeks without eating. But once I used the acclimation box method, I usually can get it to eat within a week.
I think your challenge now would be getting the fish out(?)