No flips for you!
Space is BIG. This asteroid moving at 50km/s still takes two hours to go between the moon's orbit and Earth. You could slingshot the moon to come up behind a la Armegeddom but flip the expectation by shoving it past the Earth rather than blowing it up.another wrinkle in the heroics is catching up to this thing and matching speed to then get behind it to push.
that seems like it could be action packed, but also complicated. unlike superman, where smashing headlong into it would be easy, if she flies toward it, she's got momentum in the wrong direction once she gets there. I assume she could zip out to some midpoint, then reverse direction and bring herself back up to speed as the asteroid catches up
I'm sure the mass-driver system she uses defies conventional physics, but...
By the by, the close pass to Earth would likely fling the asterood out of a passing orbit, so no need to destroy it before it comes back -- it most likely won't. An easy, and not unlikely, out is that it's thrown into the Sun after the near miss.