IIRC, the trait you are referring to is the gear driven style of LSD (Suretrack, Quaife, Helical) & not the clutch type (Cusco, KAAZ, ours).
It sounds like your rear plated diff also needs some shims!

Gear type LSD's (helical, torsen, quaife) do exhibit this yes. The suretrack are viscous, so rely on differences in speed to lock, and therefore won't do this.

Clutch type diffs like ours will do this if there is little/no pre-load (static friction). This does not mean that the diff won't operate correctly when both wheels have (well, at least a little bit of) traction though.

Adding more preload will return the low load/traction loackability of the diff. I guess you could shim the springs or you could also add springs with more force?

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Even a shimmed up plate diff will only have ~100nm of preload, not enough to drive a car uphill on one wheel. Put some lock on the centre diff and apply some brake so the wheel in the air can't spin excessively. Of just get more of a run up and don't stop at the point where the tyre comes off the ground.

I fitted the two 0.25mm shims (I had them laser cut) on either side of the pressure ring and OMFG!
I probably should have only used one shim, as the it's like I have a welded diff! :-/
With one rear wheel off the ground, I can not for the life of me turn the other wheel! I can get up my driveway piece of cake now!
Making 90 degree or U turns makes the inside wheel skip like a bugger!
Will see if it loosens up over the next few hundred KM's but I think I may have to strip it again to remove a shim!
Thanks for the help, fellas!