What vanilla animation would be the closest to something like throwing a ball up into the air?

I am making a mod that allows followers to toss bottles up into the air for you to shoot with your bow for training.

Perhaps IdleWave (IDLE form with the FormID 0003EA32 in Skyrim.esm)? You'll have to stop the animation before the NPC starts the actual waving motion by forcing IdleStop_Loose (0010D9EE in Skyrim.esm) or another similar animation that can be used to stop other animations.

In case you were not aware: you can preview animations in the Actor window. Make sure to have "Preview: Full" checked and then click on an animation in the list. Some animations won't play, possibly due to conditions that have to be met or because the animations belong to a behavior file that is not applicable to the actor.