I think kerosene or varsol [paint thinner would be OK. Then blow out with air. I guess the safest would be to pull the outer hubs and then clean the carcass. Possibly a garden sprayer would be usefull. Interesting piece of machinery. I needed ring and pinion for my former 1936 series 40 Buick coupe. One of the differential carrier bearings let go and your unit looks almost exactly the same. I got a used rear end, new bearings [single race ball bearings] and had a shop changeover the gears and put in the new bearings. They used colored lead powder to check for the proper mesh pattern of the teeth. I'm not sure about preload on the bearings or the gear backlash but specs must be available. Having said that, if a shop charges 3-5 hours as estimated by g then the price difference is not that excessive. I once had a '73 Saab and moved the ring & pinion .003 thousands closer together in an effort to reduce gear backlash which seemed excessive. 30,000 miles later the pinion gear sheared off its shaft. Were you doing something that loaded the diff unit which caused the failure. You might also pm Selby on this site since he went thru a diff change also. I appreciate you wanting to keep your own parts in there, rather then someone else's part that has been rebuilt so good luck with your quest and let us know of your progress.