A Starship sized vehicle using the well tested NERVA engine would be able to add around 20 km/sec to the velocity it already has in earth orbit. So pretty close and gravity assists would increase that.

Once you dock with Oumuamu, you can run it’s material through at an 890 ISP and gradually accelerate to much higher speeds.

You underestimate the problem.

First you have to add something like 40 km/sec to go from Earth orbit to escape into interstellar space, and then you have to add another 20 km/sec to match speeds with Oumuanu. That's 60 km/sec, which is 3 times the speed and takes 9 times the energy.

Pioneer and Voyager got to interstellar space with gravity assists. But those work best when you barely get to a planet's orbit, then it picks you up and you go close to 180 around it. (Thereby adding 2x its velocity to your own.) Once you start getting to the velocities needed for Oumuamu, you just shoot by the planet and don't get much of an assist. (Assuming, that is, that there is an alignment between the planets and a random interstellar visitor. Which is extremely unlikely.)

I stand by my statement. We do not have good enough rockets for this maneuver.