There are many ways to bypass collecting the Fairy Slingshot as a child. Using a few speed tricks, you can finish Inside the Deku Tree without ever collecting the Slingshot. Various other sequence breaks allow you to skip the Fairy Slingshot altogether.
Inside the Deku Tree Edit
As Adult Link, you can come back to Kokiri Forest to enter the Great Deku Tree. However, the entrance to the dungeon is closed when you are an adult. To enter it, you must get into the area containing the Great Deku Tree without loading it.
Method 1 Edit
Method 2 Edit
Once inside, you can easily open the chest containing the Fairy Slingshot.
Bottle Adventure Edit
As with every equippable item, Adult Link can use BA to put the Fairy Slingshot onto the B Button. The Fairy Slingshot has a decimal value of 6, and also points to itself from C-Right. You can either equip the Slingshot to C-Right as Child Link when performing Bottle Adventure, or if you prefer, you can have 6 of an item (such as Deku Nuts) with the appropriate item on C-Right (such as a Fish, in the case of Deku Nuts).