Is it possible for one black hole to pull an object out of another black hole?
The question really boils down to the dynamics of event horizons when black holes merge. It turns out that there are some great simulations that explore these dynamics. If one scrolls down to the bottom of this black-holes.org page one can see a video of the merging of two different sized black holes. One can review the underlying paper and see that the actual development of the simulation was very extensive.
The actual event horizons do move and oscillate, so the question is whether the spaceship itself has become some sort of physical element of the black hole after it has crossed the event horizon. Since it is argued in most cases that space craft can cross the event horizon in large black holes without witnessing any sort of significant effect, although the spacecraft's mass must be considered part of the black hole's mass after crossing the event horizon, it still has some freedom of movement.
We can see from the simulation that the geodesics that define the event horizon fluctuate when the holes merge. So if the geodesics fluctuate is it possible that the spacecraft would find itself on a geodesic that suddenly allows for an escape?
The answer should be no. The geodesics defining the horizon require trajectories with velocities greater than the speed of light. The spacecraft can not exceed the speed of light. So while the geodesic its on might distort during the merger of the black hole, it is the underlying space itself that is distorting, which is not going to impart some ability to defy local laws of physics to the space craft. As such, it will stay inside the blackhole horizon since its geodesic, while distorted, will still remain inside the horizon.
An event horizon is the theoretical boundary around a black hole beyond which an observer cannot be in contact (i.e) the region from which light or any other radiation cannot (classically) escape from the inside.
In my opinion, it's NO because once an object has reached beyond the event horizon, we don't even know what happens to it inside. It can't even try to escape. Only spacetime struggles inside the blackhole because, all the possible trajectories of the object point towards the singularity. An observer would see the spacecraft getting red-shifted so much and finally disappear at the moment it nears the horizon.
Even though the spacecraft is pulled into the other black-hole during the merger or hits the singularity, we can't observe it. Instead, we can satisfy ourselves that two black-holes which are close enough could merge to form supermassive blackhole and the spacecraft is shared by both.
In the scenario you're describing, the event horizons of the black holes will merge. Once they have merged, they can't split apart again. This is a theorem proved by Hawking and Ellis as proposition 9.2.5 in The large scale structure of space-time, p. 316.