Try reverse-pickpocketing a more powerful weapon into her inventory.
Pickpocket all of her ammo, punch her, then put up your fist/weapon and she will yield and have it in her inventory.
Pickpocket her ammo and give her some bad gun and ammo for it (9mm pistol and 9mm rounds for it ) then fast travel to somewhere and then back.. she should have the shotgun in her inventory now
Too answer the actual question: Its not in her inventory because she has it equiped. Can't pickpocket equip things. To get her to un-equip it. Look above.