I have found a lot of what is said about AHB to be myth. I deal with them pretty regularly as I am close to the border. They can be a pest, but they are nowhere near the monsters that some people liken them to. More like wild, untamed, feral bees. Keep in mind, most that I have run across are hybrids of varying purity and not full Brazilian bees. My state also had it's own version of AHB in the AMI that the spaniards brought over years back, so it's nothing really new for the desert.
I have found they can easily survive winter in many places, at least the hybrids can. They tend to make more bees than honey in the swarm season. You will have entire frames of brood with a tiny little bit of honey up top. They will still make honey - a lot of honey - if you manage them right. You have to feed them boxes of drawn comb, because in my opinion, they will stop drawing comb at the second box if you don't.
They are very swarmy. That's no myth. They like a smaller hive, so they swarm a lot more. They are not all agressive, but all of them can switch moods on the drop of a hat. I have seen them survive Winter too. I had a four deep hive of them last season that was my best producing hive. Had to bust it up and requeen it when the flow ended as it got too large to be safe to work in my opinion, at least alone. Very annoying to work, very runny bees that drip off the frames when you pick them up. Hard to do anything with them. I keep bees like that away from my real bees so they won't interbreed. They are normally waiting for a queen, or DNA tests or both. I normally get them from removal jobs.
If you want to keep your bees in an area with Brazilian AHB, just make sure you re-queen with a mated queen of known origin every two generations. That way they won't back breed back to being full Brazilians. Also, don't get too worked up over genetic origin, just requeen if you have properties you don't like - like aggression, swarminess, lack of honey, etc. The African bees do have some good traits if you can get rid of the bad. You can't beat them for mite resistance.
We actually have some bees in my area, up in the mountains, that I suspect come from AMM (black bees). They are far worse in my opinion than the Africans I deal with on occasion. Those guys will eat you alive. Maybe it is because the true Brazilian AHB is quite rare in my region and what I see the most is actually AMI? I don't know?
I know Texas is mostly Brazilian AHB, and had no AMI. Their bees are a heck of a lot meaner than the ones here in New Mexico.