Easiest would be to have the rabbit think rather than talk. Just record a voice and tweek the mid high frequencies and add lots of reverb.
You could cut parts of the rabbit out in Photoshop and then use that in Character Animator to quickly animate it talking (and moving).
Alternatively, you could animate it a little more by hand in After Effects. But AE is a very complicated program with a fairly steep learning curve.
There are tools that will help you animate the mouth of the bunny even if it's video. Auto Lip-Sync
The problem that I see is that you didn't ask the right question. The question maybe should be which tool can I learn to use in a reasonable amount of time to make the bunny talk? It wold really help us give you advice if we knew how much time you have to devote to the project.
AE can certainly do the job but the learning curve is very shallow. It takes several hours just to figure out the basics. Animation, even with automated tools lip sync is a slow and tedious process. Load me up with some video of a bunny and a 10 minute story about my daughter and ask me to make the bunny tell the story and I'll need at about a week. If I really simplify things maybe I can get the job done in three days...
If you decide to do it in Photoshop and utilize Character Animator with stills you'll need Photoshop Skills and a bunch of time to figure that out. Character animator using stills would be an easier way to do this.