Due to the large size difference between a cockatoo and a finch, your cockatoo could easily hurt the finches without meaning to, no matter how sweet or friendly he is, so first things first, you should never let them out with the cockatoo. Now, if you're going to have them in a cage inside the aviary, that could work, but the main problem I'm seeing is in the size of the cage bars. Most finch cages are going to be designed with thin bars and small bar spacing, no more than 5/8 of an inch. Any bigger than that, and the finches run a chance of getting stuck in between the bars. Now, for a cockatoo, the bars to a normal cage are going to be far thicker, with the spacing likely around 3/4 of an inch. This is because, as a much larger and stronger bird, he could easily bend and/or break the bars of a cage meant for finches.
With that being said, if you want to keep the finches inside the same enclosure as the cockatoo, you are not going to be able to simply buy a cage, you will have to make one. It will have to be both strong enough to keep the cockatoo from being able to break or bend it, and small enough to keep the finches from being injured. You'll also want a double layer, so if the cockatoo starts climbing on the outside, his feet will not be able to reach inside the finch cage.
In short, you have to make sure that there is no possible way that the cockatoo can get to the finches or break any part of their cage. In the long run, it would likely be easier to just keep the finches outside of the aviary.