It's very similar to the question noted in the comments which has a nice mathematical answer.
But some comments in your question are incorrect and worth addressing.
As far as I can see, the escaping star can at most end up with the
kinetic energy of the other star on top of its own, accelerating it by
a factor of 2–√ if the stars have the same mass.
The combined kinetic energy remains unchanged, unless there's a collision or some kind of relativistic interaction, but that's not really important. It's easier to think of a gravitational slingshot interaction between a larger object (like Jupiter) and a smaller object like a Voyager spacecraft. The kinetic energy and momentum exchanged is equal between the two, but the acceleration to the larger object is negligible but the acceleration to the smaller object can be up to 2 times the velocity of the larger one plus the initial velocity, or 2U + V, which is more than enough to escape both the central black hole and the entire galaxy when the initial velocities are high enough.
See this question for more math.
And you wrote this:
A "slingshot effect" is impossible because the central black hole is
stationary relative to the galaxy.
This isn't relevant, because it's not the central object's velocity that matters but the velocity of the stars orbiting the central black hole, which can be very high. When two stars orbiting the central black hole pass close to one another, then a gravity assist happens and the two stars exchange velocity and momentum.
This can be a hard thing to picture. It's often described as bouncing a ball off a moving train, the train's velocity (times 2) can be added to the ball's initial velocity. The problem with that image is that there's a point of impact where velocity is reflected and that moment never happens in an orbit. The way I like to think about it, is, imagine you're standing on a station and a train is approaching and the train is carrying a very heavy object - say a neutron star, and you throw a ball towards, but just behind the neutron star.
You throw the ball at 100 kph and the train is moving towards the station at 100 kph. The neutron star on the train sees the ball approaching it at 200 kph, it then swings around the Neutron star's gravity and flys away from the Neutron star at 200 kph, because orbital speeds are conserved in the hyperbolic orbit.
But if the ball is now flying away from the train at 200 kph and the train is moving towards you at 100 kph, the ball, inevitably is traveling in your direction at 300 kph, or 2U + V.
Does that mean that this effect necessarily involves binary systems
with very different masses so that more energy is available for
transfer to the lighter star? Or are there other effects at play?
This is an interesting question and the math is a bit messy so maybe somebody else could do it. Two stars of equal mass and opposite directions could accelerate each other and one would fly away from the black hole and the other would fly towards the black hole. I think it's easier with a larger and smaller object where the smaller objects is accelerated and the larger one only moved a tiny bit, but I see no reason why two equal mass objects couldn't exchange significant velocity to each other, changing both trajectories. That's a good question whether that would be sufficient to achieve escape velocity for one of them.