Haste Spell: "That action can be used only to take the Attack (one weapon attack only), Dash, Disengage, Hide, or Use an Object action. "
- It would have to be the attack or use object action for haste.
SSI: "While holding the object, a creature can take an action to produce the spell’s effect from it, using your spellcasting ability modifier. "
- It's an object interaction per Spell Storing Item.
DMG p.141 "If an item requires an action to activate, that action isn't a function of the Use an Item action, so a feature such as the rogue's Fast Hands can't be used to activate the item."
- Because its an object interaction that uses an action its disqualified form the "use an object" from the PHB p193. I don't think "use an object action" is different from "Use an Item action". I am pretty sure its the same thing but just written at different times and they missed lining up the language of item and object. Which I can confirm because while the item interaction paragraph in the DMG says "Item" the example in the PHB of fast hand's uses "object"
Fast Hands (PHB .p97)
Starting at 3rd level, you can use the bonus action granted by your Cunning A ction to make a Dexterity (Sleight of Hand) check, use your thieves’ tools to disarm a trap or open a lock, or take the Use an Object action.