Breeding out such an unvariable trait would be very difficult. When was the last time you saw a bee without a stinger? Now, find two in two different hives that contain such a bee and select a queen and a drone from those hives and mate them. Maybe you'll get a couple bees without stingers, maybe not.
And what makes you think lacking a stinger will be a survival advantage? Let's say you manage to breed a race of stingless AHB and loose them into the environment. What gives them better survival odds that allows them to displace the existing AHB? The fact is, without stingers, they are probably less fit to survive and won't really displace any sting-equipped AHB.
AHB is going to be hard to breed out because it is partly the "bad" traits that make it more successful. So, long as there are feral bees, they are going to have an advantage with AHB genes.