Why did Edmund Barton carry around a gun?
There is no one gun that is gender exclusive. For any person, male or female, the best gun to carry is a gun that fits the person's hands properly, and that fits the person's carry requirements. Some people want a tiny pocket gun, some women want a gun that can be dropped into a purse, some women will actually carry a gun in a holster, so, in short, it depends on the person.
A BAU agent is allowed to carry a gun as long as they have passed their test to be eligible to carry and shoot a gun. To be able to carry a gun you must know how to shoot and hold the gun steady. More correctly, you have to be able to pass the FBI shooting qualification test, and that applies to all agents, not just BAU.