Are you saying if the rearwheels are on the ground then the handbrake works fine?
Handbrake drum is in between the rear driveshaft & gearbox.
If there is no handbrake at all, you should immediately address it by adjusting the handbrake.
Even with the LSD arrangement it makes no sense. Once the rear driveshaft is locked by handbrake, the pinion gear can't move and then the crownwheel can't move. That means wheels can't move regardless of what spider gears can do. This is true regardless of whether you lift one wheel or both rear wheels up.
What I dunno is whether GUs have the handbrake arrangement somewhere else other than the above location between the driveshaft & gearbox. But, I seriously doubt that makes any difference at all.
Or whether it has some clever arrangement to cancel the handbrake when rear wheels are off the ground and transfer the handbrake to front wheels. That would be something I would love to see.