We do this fairly regularly with space probes already.

https://en.wikipedia.org/wiki/Gravity_assist

You steal energy from the object you're slingshotting off. The black holes are slowing (infinitesimally) down every time you do it, gifting their energy to the accelerated object.

Hence the article talking about potentially detecting this sort of behavior by monitoring for prematurely merging and overly eccentric binary black holes.

The paper says that the maximum velocity obtainable would be 2x the rotational speed of the body used. Some such bodies rotate at appreciable portions of the speed of light.

Obviously you cannot exceed the speed of light but it doesn't seem like it would be a wash.

At some level, that's also what glider pilots do to take off from mountains.