If you open up the diff you'll find it's basically 3 parts -
1) the input drive pinion (leave this alone)
2) the crown wheel
3) the "centre"
The crown wheel bolts to the centre, whilst the centre contains all the LSD gubbins. So....
If you get a good LSD from a scrappy but the gear ratio is wrong, then you can just open up your existing diff, whip out the centre+crown wheel, swap the crown wheel onto the centre from the good diff, and re-assemble. This means you're using your old pinion and crown wheel (so the gear ratio is the same) but a new LSD centre.
Obviously there is a bit more to it than that, and you must ensure the backlash and side bearing preloads are correct on re-assembly.
Disclaimer - I've not done an Omega diff but I am in the middle of doing a Lotus Carlton one (which uses cones not plates) so the internals are similar. Hopefully it'll be going on the car this weekend.
https://www.aussiev8.com.au/driveline-suspension/71923-btr-m78-diff-question.html