Remember, in 2e, when you put inferior armor over superior natural armor (or layer superior armor over inferior natural armor), the result is that you improve the AC by 1.

So, if you put leather armor (AC 8) on something with a tough hide (AC 9), you wind up with an AC of 7. If you put chain barding (AC 5) on something with an AC of 7, you get an AC of 4.

So, any kind of effective armor on the bird will improve its AC by at least 1.

You can also rule that some measure of its 4 AC comes from dexterity. If you say that it comes entirely from their dexterity, then you apply that on top of the AC from the armor.

Cheap, easy method, though? Armor spell. Ring of Protection is more expensive and less effective, but longer lasting.