So I have seen on in several places that it is possible to pickpocket trainers after training with them. Is there a list of pickpocketable trainers? I tried to do this with Babette in the Dark Brotherhood... but when I go into sneak mode... it seems I am unable to pickpocket her as when I place the crosshairs over her it shows nothing. Are there only certain trainers this works with?
Children are not valid pick-pocket targets in Skyrim - Babette isn't pick-pocketable because she's a child vampire, not because she's a trainer per se.
Pick any other trainer, and you'll be able to recoup your expenses.