My lyretails and barlets eat flakes. Pictilis (a large anthias like squareback) sometimes eat flakes but mostly prefer mysis.
In your case, those are 3 anthias species that have significant size difference. With that much of size difference, it is hard to find a single type of food all 3 would eat.
Lyretails and resplendent might eat calan-eeze. For squareback, maybe freeze dried mysis. You can try mixing these two food and use a auto-feeder. But the issue is, these foods would not readily sink. They will just float over the water and will be removed by the overflow. You can try to autofeed to the return pump section of the sump. That way the food will be blown to the tank when it sinks. Or you can use a feeding ring.
Also why do you need an autofeeder. You can feed 3 times a day (even 2 after anthias get used to eating bigger meals) and that would be sufficient. I used to feed the tank in the morning before I leave for work, one time more after I arrive and one more time 10-15 mins before the light turn off. After 3 months or so, I reduced two feedings in the evenings to one and anthias is okay with it. Pictilis actually only eats once a day (which is relatively okay for larger anthias species).