I changed my diff fluid and I never smelled such foul odors. Do not spill it in the garage! 100,000 miles and 15 years old? Who knows. The Prev owner had no records and I don't know if it was ever done.
I wondered if it is done on the flat floor. I jacked up the rear and filled the diff then set it down, let it drip and then jacked it back up and put in the plug. I think that works. I used regular gear oil but added some additive for the limited slip diff.
Seems to be great 15,000 miles later with no noises or anything.
1985 Mercedes 500SL Euro (Gray market)
1995 BMW 520i Euro (Gray market)
1992 BMW 525it Wagon
1994 Honda Del Sol Si