You've eliminated some things. Since they act sick before they die it is not some type of predator and they are not getting killed because they pile up. It's not one chick killing them, I've had that happen but they were 2 weeks old. You don't report any wounds.
It would be interesting to see a photo of the brooder, mainly what kind of ventilation it has and how those heat lamps are set up. In those temperatures I use two heat lamps (the second one mainly for redundancy) to heat an area in my 3' x 6' brooder. That's a fairly small area out of the entire brooder but I have great ventilation up high so the rest of the brooder cools off quite well. You did not answer how cool the rest of the brooder is away from the heat lamps but with much ventilation I'd think it would be hard to overheat all of a brooder that size. They should just move to an area where they are comfortable. One thought, how warm is the weather when this happens? One challenge to brooding outside is the temperature swings. You don't want to overheat them in warmer weather.
At 5 days old it would be surprising if it were Coccidiosis unless the brooder is wet and it wasn't cleaned since the last batch was raised in there. Having some of the bugs that causes Coccidiosis in their system isn't a problem until those numbers get out of hand. Due to the life cycle of that bug 5 days should not be long enough for those numbers to build up enough to cause a problem. It won't hurt to treat them all with Corid for Coccidiosis but I really don't expect that to solve the problem.
It's the bantams, not the larger chicks. That does not sound like a disease or poisoning. After a chick has been dead a while it tends to flatten out but not immediately. That's kind of mystifying. And they are scattered around, not in one spot. That sounds like they are not piling up to stay warm or getting as far from the heat as they can, so the heat doesn't sound like it is that far off. Sometimes when they panic they can pile up in a corner and some get trampled, but it sounds like they are not in a corner. Besides, they act sick before they die. I'm not sure what that leaves.
A chick can live for well over three days just from absorbing the yolk as it hatches. I don't know how old they actually were when you got them. I had a chick that did not learn to drink when I dipped its beak or from watching the others. It was standing by itself giving a plaintive peep, you knew something was wrong when you heard that sound. I dipped it's beak in the water again and it got the message. It stood there and jut kept on drinking. That was about 5 days after hatch. That does not sound like what you are describing.
Why that chick perked up when you took it inside but died when you put it back is also hard to understand. What did you do to that chick when it was inside? Did you warm it up or maybe feed or water it?
With it being the bantams and not the others (unless the ones in this brooder are all bantams) it seems to come down to them being smaller than the others. I would not expect them to be bullied because of size, many bantams are so feisty they bully larger chickens.
Nothing jumps out as an obvious problem. I'd look at how hot and cold the different areas of that brooder are in hot and cold weather. I'd make sure they had room to eat and drink. If one is acting sick I'd give it some energy water. That could be electrolytes or sugar water, I've used hummingbird liquid. Put a drop of liquid on the tip of its beak and let it swallow that. Do not force liquid down its throat, you can drown it. I just don't feel good about any of this, nothing stands out. Maybe it is just coincidence that you are losing so many.
