Well, I was testing this Question with Mew on Pokemon Showdown, and I saw that Sturdy, at times, was not activating before Focus Band (I was using Level One Pokemons). Which go me thinking...
Do Items have a certain priority, just like moves? If so... Are there specific items that will activate before an Ability?
Quick Claw is the only Item that activates before an ability, but no, Items do not have a Priority.
Source1 and Source2