2. Kate's would be the most natural answer.
1. This would make sense to an English speaker but it doesn't quite sound natural. For me, it sounds old fashioned, same with number 3.
Also with number 1 and 3, it is unnecessary to state that Kate has apples because just by saying 'Kate's' it is automatically assumed that Kate too owns apples that are not as delicious. 'Kate's', with the apostrophe ('), is the reason for this assumption as it gives ownership to the apples.
4. This is closer to the correct answer and once again can be correct but is not as natural.