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I haven't had the time to sort out the various arguments presented, but I
find the concept of a single stationary charge in a gravitational field
spontaneously radiating energy rather unnerving.
I note that the original question did not specify uniform
acceleration or even straight-line acceleration
As for the case of hyperbolic motion, I will need to think
about that some more. I would expect classical electrodynamics
to have an unambiguous prediction, one way or the other.