I've run into similar issues. I think it might be precisely because it's awkward in English. Technically correct, on a word-for-word basis, but it doesn't reflect a faithful translation. I've learned a lot from what Duo will and won't accept, and I think that's how we grow in our study of language.
Word for word translation isn't really useful when you go against phraseology in either language. In a sense a duo (or trio, etc, lingo approach is to make speakers and listeners able to automatically do the inversions which might be needed for translating one language to another. If you wanted to say to a French person 'It's the children's rabbit, you'd need to have internalized the grammatical shift. So Duo presents is (more or less) with what makes sense in each language
If you heard "c'est", you know that it refers to a singular noun (le lapin). Remember that "ce sont" (they are) is the plural of "c'est" (it is). Ce sont cannot be "it is". The problem introduced by mistaking "leur" (possessive word) for "le" (definite article) is that you already have a possessive phrase that comes after, i.e., (le) lapin des enfants. The result does not make any sense in French and so could not be coherently translated to English. Try to understand the whole French expression first and then translate it to English.
Please refer to the comment from C.J.Dennis above. "Des" in this context the way to form a possessive when the owner is plural, i.e., le lapin des enfants = the children's rabbit, literally "the rabbit of the children". In English, the standard way of expressing it is "the children's rabbit". Some stop short of a full translation and are left with the word-for-word version, "the rabbit of the children". It is technically not wrong, but it is not the most common English either.
There are different uses for the French "des". One is (as you say) the plural of "un/une". The other is used with plural possessive forms, as in this sentence, "c'est le lapin des enfants". Do not confuse these two different uses of "des".