ypur LC probably already has a LSD diff.
you can test by jacking up the back axle to lift both back wheels off ground.
chock front wheels
make sure handbrake off, gearbox in neutral and transfer
and turn one back wheel by hand, if the wheel on opposite end of axle rotates the same direction you have an LSD diff,
if it rotates the opposite direction, its an open diff
but a true locker usual performs better and the toyota LSD clutch plates are often shot by 60k miles, so the LSD doesnt work anymore