I'm not sure how the RAW should be read, but it does seem to make the item significantly better than it was intended to be.
There is precedent which allows earlier access to class features using similar language.
The robe specifically states that it grants access to powers at earlier levels: "The wearer treats her sorcerer level as 4 higher than normal for the purpose of determining what bloodline powers she can use and their effects." Since that could easily be a case of specific overriding general rules, I don't think it's a good precedent.
Your effective level for that class feature is treated as 4 levels higher, but your BAB (which is what most of the AAT options go off IIRC) would not be altered.
Nope. Only 1 AAT option uses BAB (master armourer). 6 scale off level, and 3 don't scale.
It's a 4k item. At level 3? That's a pretty huge investment.
Level 3 is just an example. You could also be a 7th level fighter wondering if being treated as an 11th level fighter for Armour Training means that you get access to your 11th level AAT pick.
If a 7th level fighter with Armour Specialization wears a Sash of the War Champion, he gets +1 AC for treating his fighter level as 11.
If he also gets to pick a new AAT as an 11th level fighter would, he can also take Armoured Juggernaut and get 1-3 points of DR/- depending on type of armour worn.
Bravery also increases from +2 to +3, which could mean an extra +1 to all will saves if the character has AWT (armed bravery).
+1 AC, DR 3/- in full plate, and +1 will saves is a really good deal for 4k at level 7.