I agree with Pax. I'm new here (2nd post), but I will [humbly] try and say it in a different way.
Based upon the wording of Arcane Disciple, Bards and Sorcerers are restricted to casting a maximum of 1 domain spell per level per day. Bards and Sorcerers are classes which "cast spells without preparation". Wizards are not restricted by this - their restriction is that they can only *prepare* a maximum of 1 domain spell per level per day. Wizards dont "cast spells without preparation" -- even Wizards who have this feat. (For that matter, IF anyone who took this feat became someone who could "cast spells without preparation", then there would be no need for the writers to clarify it - the special relative clause would simply apply to everyone who had the feat.)
For this arguement, you don't need to consider the SH feat at all. You simply need to decide whether taking AD suddenly changes the character into someone who can "cast spells without preparation".
Based upon the wording of Arcane Disciple, Bards and Sorcerers are restricted to casting a maximum of 1 domain spell per level per day. Bards and Sorcerers are classes which "cast spells without preparation". Wizards are not restricted by this - their restriction is that they can only *prepare* a maximum of 1 domain spell per level per day. Wizards dont "cast spells without preparation" -- even Wizards who have this feat. (For that matter, IF anyone who took this feat became someone who could "cast spells without preparation", then there would be no need for the writers to clarify it - the special relative clause would simply apply to everyone who had the feat.)
For this arguement, you don't need to consider the SH feat at all. You simply need to decide whether taking AD suddenly changes the character into someone who can "cast spells without preparation".