I would be interested to no how this turns out. As I bought the right type 3 times the 36:11 and they all made terrible noise. Got the job down to 40 minutes to get it out in the end without even taking the wheels off. Your new one 38 Crown 13 spigot should bring the revs down.
I have been chasing a "Sonic Boom" at 80mph on power for 3 years with mine, Diff swap 6 times. Changed prop bearings first then the whole prop.Complete exhaust. CAT joints,turbo clamps. Engine mounts,Gearbox mounts. Added the Harmonic balancer to the the rear diff. Gearbox oil and filter made no difference.
I noticed a 400rpm drop before when when I up sized the tyres so hopefully your new diff will do the same and be silent. Rebuilds on these diffs are limited to bearings only as nothing else is available (like worn out planet gears) very visible with the the cover off.Also the large outer driveshaft bearings move in and are not held with a brace over clamp.
Having to STAR it up and find its a rumbler would be very disappointing.So hope all goes well. Your choice of oil may play a part latter on if its hot there.
I had to use a digital lazer temp in the end to find out if the diff was anygood and not suddenly lock up at 70mph which is scary as hell believe me(like someone stamping on the hand brake!!). MB oil is very thin for better MPG and in someway gets to the parts faster for lubrication. But if the diffs done a 100k its only got 30k left. Tried loads of Oils and settled straight 140. Thick as hunny and takes 5 miles to thin down but at least its 45-55c instead of burning your hands to touch the diff 75c. Used a coke bottle with compressed air to get it in" Its thick.