The paradox is, sort of, resolved as follows: the number of photons changes when you switch between non-inertial frames. This is actually a remarkable fact, and holds also for quantum particles, which can be created in pairs-antipairs, and whose number depends on the frame of reference.
Now, a step back. Forget about gravity for a moment, as it is irrelevant here (we are still in GR, though). Imagine a point charge, which is accelerating with respect to a flat empty space. If you switch to the rest frame of the charge, you observe a constant electric field. When you switch back to the inertial frame, you see the field changing with time at each point. This naturally corresponds to appearing magnetic fields, and hence radiation.
In the presence of gravity the case is absolutely similar. To conclude, switching between non-inertial frames makes a static electric field vary and hence represent a radiation flow.
Another point might be: When moving with charge, no energy is emitted, but when standing in the lab frame, there is a flux observed. However, there is no contradiction here as well, as the energy as a quantity is not defined for noninertial frames.