Public information on rabbits in space is surprisingly sparse. The earliest instance I have been able to find is the July 2, 1959 launch of the Soviet R-2, which reportedly carried the first rabbit into space.
As mentioned in the question, a rabbit was also aboard Korabl-Sputnik 2, which is more well-known for the dogs Belka and Strelka.
I did find a 1999 Japanese study, Development of the aortic baroreflex system under conditions of microgravity, which refers to science performed on rabbits in microgravity. Unfortunately, it is only the abstract, and I was unable to find which mission this was associated with.
That's about it.
Searching the NASA domain, I found reference of lots of incidents of rabbits in space, but none with any detail. I did find one old document (which I've since lost) that mentioned that one of the cons of using rabbits for space experiments was their high susceptibility to stress.
I found another Japanese paper, Life science experiments performed in space in the ISS/Kibo facility and future research plans, containing proposals for experiments aboard the ISS through 2024. It has one brief mention of rabbits:
Mammals such as mice and small rabbits are to be grown and bred in space to analyze fertilization, growth, development, movement and living under µg and 1 g conditions. Centrifugation systems in the ISS will be used to provide different levels of gravity. Potential projects are studies of reproduction, development, birth, care of offspring, behavior and healthcare in gene knockout (mutated) and parental wild-type animals. Multichannel telemetry will be used to monitor body temperatures and behavior with a CCD camera and to download data to the Earth from the ISS.
I speculated in the comments that if they keep sending rabbits up, one might expect them to have at least some survivability. On the other hand, perhaps people just really want rabbits in space.
Also in the comments, Chad pointed out that "[a] living and stressed rabbit will generally not eat or eat very little and will not reproduce" and a 1g centrifuge for rabbits would "require [a lot] of equipment for a few animals on the ISS".