The energy goes into heating up the star, not into motion. The star will get very hot, perhaps even explode, but it won't "bounce". Remember, the small object doesn't fall into the big one, they both fall into each other, so the there is no net creation of momentum. It all cancels out.

Let's suppose the two gravitationally interact so that they fall onto a common center of gravity. As they fall, they pick up speed due to the conversion of potential energy relative to each other to kinetic energy. So, how much of that PE is available to convert to KE? The momentum equation tells you, the resulting combined body has AT MOST the same momentum as the two bodies did when they were separate which is 0. All momentum gained by the fall is cancelled out because they are in opposite directions.

I.E., the fact that the two bodies fell towards each other may have liberated an enormous amount of potential energy, but since the momentum must be the same after the collision as before, and since the bodies "stick" to each other, the resulting body is going nowhere. This energy must go into deformation and frictionj of the two bodies in as they collide.

This is a different situation from a collision in which gravity plays a negligible role. For example, out in space, if you throw a 1 kg brick at a 100 kg man in a spacesuit at a speed of 1 m/s the momentum before the collision is 0 for the guy in the suit + 1 kg-m/s for the brick. So the total momentum is (100 x 0) + (1 x 1) = 1. After the guy catches the brick, he now weighs 101 kg so he must now be traveling at a speed which still yields 1 kg-m/s total momentum.

I.e, mv = 101(v) = 1
v = 1/101 m/s he's going slow, but he's heavy, the mv is the same.

However, if the brick had not been thrown, but the man and brick had collided due to the mutual attraction of gravity, the total amount of momentum of the entire system would have been 0, before and after the collision. They would have met in the middle and just stopped.

I suppose it is possible for a star to pick up some velocity by colliding with another object, but the momentum of both objects must be the same before and after the collision (for an elastic collision, like billiard balls). In other words, the velocity of collision due to their mutual gravitational attraction cancels out. Only any velocity relative to each other they had prior to their interaction will be available for momentum exchange. With objects like stars and planets, there's going to be an inelastic collision with extreme deformation, so much of that "momentum" will be lost to the particles and the fragments and the heat.