The mass of such an object is likely be miniscule compared to the mass of the earth. So even a very close approach isn't likely to make much difference to the earth's orbit.
E.g. an asteroid 100m across with the same density as the earth is going to have ~0.0000000000001 the mass of the earth.
Mass is relative to speed. If the object has a very high speed, it can reach the mass of the earth or even higher. That's why my understanding is that if it can escape the gravity of the earth, it means its own gravitational field is as or more powerful.
The mass of an object increases appreciably with velocity only at relativistic velocities. This is completely irrelevant at the speed an asteroid is likely to be travelling.
If an asteroid travelling at relativistic velocity passed through the atmosphere, a change in the earth's orbit would be the least of our worries!
E.g. an asteroid 100m across with the same density as the earth is going to have ~0.0000000000001 the mass of the earth.