Ok, i've realized why the increase in mass is important here. The added mass does not have any momentum in the current direction of the satellite, and since momentum is conserved the satellite will slow down, lowering its apoapsis and, if you repeat the process, changes its orbit closer and closer to a circular one. So any energy "extracted" will be taken from the inner satellite's orbit(?).

This did lead me to two different questions though. The first is a change of setup. Send your light beams this way instead:

In this case the momentum of the light will add/subtract to complement A's orbit. Simplest calculations would put the momentum of the satellite at p=mv, or v=p/m. The momentum added by the light will be p

_{l}=E/c, and the mass it adds is m

_{l}=E/c

^{2}. So the change in the satellite velocity is Δv = (p+E/c)/(m+E/c

^{2}) - p/m. Right? Now, it seems to me like this is positive because the numerator of the first term increases more than the denominator since E/c is greater than E/c

^{2}. Or in other words, the light adds more momentum than it adds mass. In that case, what if we beam some light from this direction and some light from the opposite direction to keep the satellite velocity constant. Won't its orbit then stay the same and the original problem persists?

However! You don't add momenta linearly in relativity, do you... How do you add them? I don't know, but if Δv ends up less than 0 there should be no way to maintain the satellite orbit.

The second question i thought of is more general. With this new setup the light won't fall straight, will it? It will be curved by the gravity well. At that point it starts to look like simply dropping an object, so my questionn is this:

Is gravitational blueshift analogous to the kinetic energy gained by a falling mass? My rationale for why very strong gravity is needed for noticable frequency shift would be that light is already all momentum, so you need

a lot of change in potential energy to significantly change its momentum. Matter, on the other hand, can easily have zero momentum, so any energy added will quickly make a significant difference.

Is that at all close to true or is there more to it than that?