Answer
As the car accelerates forward, the air and her head tend to stay in place (Newton's first law).
The effect on the air is to compress it in the back of the cabin, which increases the air density and the pressure there (page 270). The increased air pressure in the back of the cabin pushes the balloon forward. Combined with the effect of the normal, upward buoyant force, the balloon leans forward.
Note: There is an inertial effect that would tend to make the balloon snap back just like the passenger's head and the air, but that effect is overcome by the force due to the horizontal pressure gradient.
