If you have a hoist, I suggest removing the back plate and visually checking the gears.
Certainly check that the grease nipples at the outer ends of the axle have been used. The grease serves two purposes: to grease the wheel bearings but also to stop the oil seeping past the felt seal.
A word of warning about changing to a BTR axle - you may end up having to change the hubs, brake back plates, shoes and drums; this could escalate the cost considerably.
My recommendation is to do the visual check and, if there's nothing obvious, change the oil and grease the wheel bearings. There is a very remote possibility that the LSD plates have gummed up and are stuck with lack of use.
If that doesn't work, consider removing the axle and remove the diff (whoever does this job will need to spread the diff housing a tad as the diff is slightly larger than the hole in the housing) and despatch the diff unit to Billy Bellinger or Quafe for a rebuild. Reassemble in Oz but ensure that you've the parts required such as the collapsable nose spacer and sufficient shims to get the backlash correct.
