I've found a way to avoid most of the mourning after a small pet dies. I replace them after a week or so. They live about 2 weeks. It had been quite a while since getting my Chinchilla, and since I try to only keep rare pets, I hadn't seen a new one in a while. I finally found one. One Sim took the old one out of the cage and the other put the new one in. Ta da. No mourning. I then sold the old one. A few Sim hours later, I noticed my Sim still had that Chinchilla in their inventory. I thought I'd deleted that one, so I got rid of it again. Ooops. That wasn't the old one, it was the new one. I realized it the next day when I noticed the Chinchilla cage was empty. They must have taken it out to play with it and didn't put it back. I wish they would put them back. How long will it take me to find another?
edit - Update
Oh wow. That didn't take long at all. I switched a Sims 'Collection Helper' to Rodents, zoomed out, and zoomed in on the first rodent I saw. It was a Chinchilla. I didn't take any chances. NRass / Debug Enabler / Options: Chinchilla / Debug: Force Capture. I didn't consider that cheating, because I was only replacing what I already had.