There is no particular reason to believe that the awaken spell changes the animals genes.
GM could make a call either way and/or vary it with setting, but by default I would say since it doesn't mention anything about offspring, it has no effect on offspring.
If awaken is a genetic change, then an awakened animal probably could not mate with a non-awakened one, and you would have to have the 2 awakened as you proposed to get any offspring at all. Essentially then you would have created a new species entirely.
Ask your DM. I'm inclined to say you get normal rabbits. The awaken spell only grants what it says it grants (3d6 INT, +1d3 Cha, 2x d8 HD) and a change to type "magical beast". Consider it like a brain transplant into a rabbit.
If you wanted to research a variant that made those granted effects inheritable, it's probably possible.
EDIT: ninja'd
My inclination - to remove all headaches - is to say awakened animals are sterile, but that's just me.
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Well, the question was for two awakened animals; one male and one female. Since both are magical beasts of the same type, and the spell does not specify sterility, they should be capable of breeding normally within their type. Their offspring should be magical beasts of the same type since the spell makes a permanent change. Though that would create some issues of its own. In-breading would become quite an issue very quickly.
Edit: this does seem to be the kind of thing the gods would slap you down for though.
You would get bunnies lounging all over your palatial mansion, Hugh.
In somewhat more seriousness, if your campaign actually spans game years, you could do some form of selective breeding project to see what sort of gifted or twisted progeny comes from mixing animal forms and human forms.
But your character would have to move to an island, and henceforth be named "Doctor Moreau".