I suspect that it is necessary to come to a compromise that does not involve euthanasia. This situation could be resolved by providing the cats with their own building, such as one of the portable yard buildings sold most places, and then thoroughly blocking entrances to the house. The dander concentration in an enclosed space is usually much worse than it is outdoors. Your brother might even consider the option of spaying the cats and letting them stay outside and establish a territory. This would keep the number of new cats to a minimum and it seems reasonable safe to do this in Arizona, at least from a climate standpoint.
Most cats are pretty resistant to heat related problems.
One problem with this approach is that the cats will be able to spread the leukemia to other cats in the neighborhood. Another ethical problem. Perhaps he or she will have some advice on living with this circumstance. Wetting the cats thoroughly with water once or twice a week will keep the dander down and can help with allergies quite a bit but requires cooperation on the part of the cats. If your brother would consent to it, there is a chance that psychiatric care might help.