The objections that the earth should take up some of the momentum and energy are entirely wrong, since I am using the frame of reference where the earth is stationary.
Not energy, at least not any important amount, just momentum.
You cannot choose absolutely any frame of reference that you like, if you want to use Newton's laws of motion in their usual form. You can choose any
inertial frame of reference.
You can choose a frame of reference in which the Earth is
initially stationary, but if the Earth's momentum changes, as it will in response to forces on the Earth, you cannot choose a frame of reference in which the Earth
remains stationary, because that frame of reference is not inertial.
Conservation of momentum does not hold in non-inertial frames of reference.
As Subduction Zone pointed out, your argument would imply, not only that the Blackbird cannot work, but also that no car can work. When a car accelerates, its momentum changes. Where is the corresponding oppositely-directed change, to satisfy conservation of momentum? It is in fact in the momentum of the Earth, massive though it is. Its large mass implies a small velocity change, but not a small momentum change. The change in momentum of the Earth equals the change in momentum of the car, just in the opposite direction.
Also see
this post of mine, which perhaps you missed in all the irrelevancy that goes on in this thread.