Could a treaty-limit anti-aircraft cruiser carry heavy armour?

Could a 10,000-ton anti-aircraft cruiser carry a full armament and five or six inches of armour?

Probably not. At that displacement it could carry very little of such heavy protection.

A fully armed anti-aircraft cruiser on a Town-class-like layout probably could not fit within 10,000 tons while carrying five or six inches of armour. At that displacement, very little of the proposed protection would remain practical. Armament, armour, machinery, and hull volume all compete for the same tonnage. Treaty limits make those tradeoffs sharper even when the desired roles are individually sensible.