In a slingshot maneuver, the vehicle approaches the planet from outside its sphere of influence on a hyperbolic trajectory, with the hyperbola’s eccentricity determining the angle between its incoming and outgoing velocity vector. From afar, it looks like it “bounced off” the planet, but it really just swung around it. Relative to the planet, energy remains constant — indeed the incoming and outgoing velocity will be the same — but relative to the Sun it will have changed a lot, because some of the planet’s kinetic energy gets transferred to the vehicle.

The downside is you can’t just park yourself around that planet waiting for the opportunity to do such a slingshot. Hyperbolic trajectories are inherently single-pass maneuvers, unlike bounded elliptical orbits. You could conceivably park in a high-energy, loosely-bound elliptical orbit, but to do this parking you still have to lose some of your energy relative to the planet in order to achieve orbital capture, and do a burn to regain that energy later. This is probably counterproductive if your ultimate goal is a solar escape trajectory.

That’s not even the most serious difficulty, though. Intercepting an extrasolar object (aside from a brief flyby, that is) requires not just achieving any solar escape trajectory, but specifically one in the same plane as the target. And even once that is achieved, an actual hyperbolic rendezvous is an extremely difficult maneuver, not to mention time-consuming: with the delta-V budget available to chemical rockets, I’d guess (without having done any calculations) that you’re looking at many decades or even centuries to complete the rendezvous.

Due to the constraints of physics and chemical rockets, spacecraft trajectories are hyper-optimized based on knowledge of the exact mission plan. I don’t think a general-purpose vehicle stationed in the outer solar system would be suitable here.

I wouldn't even consider conventional chemical rockets for such a rendez-vous. There already are plenty of other options available: ion thrusters, or the new plasma thruster NASA is currently working on, which is very promising, or a laser sail... even that dubious emDrive if it proves working. The tricky part is really what to do on arrival if we don't want to just pass by it at an enormous speed.

Thanks for the thoughtful reply. Basics of orbital mechanics (beyond KSP triviality) is near the top of my "to learn" stack.