As the orbit speeds are far below $c$, relativity doesn't matter. The fact that it is a black hole doesn't matter either-it could be a star of the same mass. The gravitational field of a spherical object is the same as a mass point at the center as long as you are outside the object. It takes a lot to change the momentum of the earth that much. The only thing I can think of to eject earth from orbit is to have another massive body come flying through to slingshot the earth away from the black hole. Does that work for you?
Added: Here is how I would think about it. You need an orbit simulator to get it really right. I believe there is a free version of Satellite Tool Kit and there must be others on the web. For simplicity, let everything happen in a plane. You need a big body, call it A, to transfer this much momentum to the earth. I would guess 30-100 earth masses, but you might wind up with more. Figure out what velocity change you want the earth to have. 921
I am hoping we can think about the two events separately and let the simulator patch them together. The first is A falling in from infinity on a parabolic trajectory. When it passes earth's orbit, it will be travelling $\sqrt 2$ times faster than earth. We get to pick the direction, which will be set by the angular momentum of the system. The second is the scattering of A and earth, which we analyze in their CM frame, ignoring the black hole. I think it will be so fast that will be OK. Decide what velocity change (as a vector) you want the earth to feel. Find a set of parameters (incoming velocity and miss distance) in the A-earth frame that results in that velocity change for earth. Now find an angle for A to cross earth's orbit that results in the proper relative velocity of A and earth in their CM and direct A to cross earth's orbit at the correct distance from earth. Try it and see if it works. You only have three parameters-the mass of A, the angular momentum of A around the black hole, and how far ahead of earth A will cross the orbit before you allow for the A-earth interaction.
Let me know how it works. No guarantees. 675