Actually, I think memyself9 has the right of it. The -et- affix in Esperanto is the equivalent to the Spanish -ito (and the English -y) which is technically a diminutive but which is quite often used for informality or cuteness instead.
When a friend calls me Miguelito, they're not implying that I'm young or small; they're simply calling me Mikey. Similarly, a little bunny would be un conejito, whereas an adult Playboy Bunny is una conejita.
So kunikleto does imply small size and/or a young age, but also has the sense of, "Aww, look at the big widdle bunny wabbit!"
My two cents; a penny after taxes.
According to this blog post, they are called kittens! http://blog.oxforddictionaries.com/2014/10/call-baby-rabbit-baby-animals/ I would say bunny was a cutter name for rabbit.
Bunching rabbits and hares together like that would be like saying that you would use the term 'homo' to refer to both humans and bonobos: they're related, but that's about it. Both rabbits and hares are lagomorphs, and both humans and bonobos are primates, but the terms aren't interchangeable.
They should accept "like" for ami. Esperanto "ami" covers liking and loving. "Mi amas vin" is not necessarily anything stronger than "I love you" (as in French, Je t'aime). Although apparently under the influence of English, there is an increasing reluctance to use ami for "like", between humans, at least.
I brought this up at our bi weekly Esperanto meeting (Montreal)..... One member said "What do you pull out of a hat... A rabbit, not a bunny." And I replied "How many chocolate Easter rabbits have you eaten." I think both Rabbit and Bunny would be correct as they refer to the same idea... An adult "Leporidae" in Latin or "Kuniklo" in Esperanto. Good thing there's only one word for this creature in Esperanto :-D