Locking hubs are not used with this arrangement. The problems with using lockers with such axles are that the drive shaft will be driven and that the side not split by the disconnect will undergo drag, especially when the differential is cold. This will cause a pull toward the side of the front axle that the differential is on. Limited slip differentials of both gear and clutch types will transfer only a fraction of the available torque to the side that is not slipping. This is not as good as a locker, but may be enough to get you through.
Many vehicles are available with limited slip differentials as a factory option.
Limited slip differentials are not suitable for use in the front axle of a vehicle with an axle disconnect. I am aware of is made by Dyneer, the company that makes the Detroit Locker. It works by having the speed difference between the wheels wedge one of the gears in a pocket, reducing the freedom of movement of that side. This transfers torque to the other wheel as long as the speed difference is large enough to keep the gear in the pocket. This type of differential requires no special service and no special lubricants.