Can I actually turn this kid invisible?

No.

In the base game there is only one spell that induces invisibility and that is targeted only at the caster. Since children cannot learn spells, nor can they be pick-pocketed (reverse or otherwise), there's no way to do this in the game.

However, if you are on PC then you can use the console to acquire the Fade Other spell which isn't available anywhere in the game. Fade Other, as the name suggests, allows you to target an NPC with the spell and they become invisible for 30 seconds. The ID for Fade Other is 00028532 so you can use the console command player.addspell 00028532 (you can omit the leading zeros) to add it to your spell book.

Here's a reference explaining the limitations of children in the game: http://en.uesp.net/wiki/Skyrim:Children

---

Update:

If you're playing the Special Edition on PS4 or Xbox One you could just create a mod which adds this spell into the game somewhere and deploy the mod to Bethesda.net. I'm unsure if a mod that does this has already been created.