If you use the Jaguar ring and pinion for the salisbury LSD, it's a straight swap. Try to do this but new parts will be scarce as rocking horse sh1t, I bet.
Keeping the Jaguar Salisbury LSD is also a good idea because the Jag stub axles are 19 spline, and its not such a bad LSD. The Jag pinion splines are also unique, so keep to a Jag gear set and no worries. Tell your builder not to rely on backlash, that he needs to check the tooth contact pattern as a final assurance.
You can also use the Dana 44 ring and pinion to 3.77, the bolts that mount the ring to the carrier need a sleeve kit because there is 1/16" difference, and some carriers need a spacer ring between the ring and carrier, I think thats what you mean by a "thick ring gear". The Dana pinion mounts a GM spline yoke, can't use the Jag one.
Auburn, for one US manufacturer, make a 19 spline LSD that mounts Dana 44 gears as standard, but the pinion takes a GM spline yoke.