How much sewage overflow occurs? If it's only a small amount, like a cup or so, then a blocked vent could be the cause.
As water runs down a drain pipe the air ahead of the water must be pushed out. If there's inadequate venting then the pressure build up in the drain can cause water to be expelled from the trap below the sink and tub. It could cause bubbling or splashing in the toilet, too.
To answer the title question "Can Air Admittance Valve under bathroom sink fix bathtub drain overflow?" -- no. An AAV allows air into the drain plumbing but does not let air out. An AAV could resolve a problem where the trap at a fixture is getting the water sucked out of it. As water runs down a drain pipe it creates suction in its wake. If there's inadequate vent or AAV then the suction literally vacuums the water out of traps.