In relativistic physics (aka, classical electrodynamics), which applies even if v~0, an accelerating free electron, in arbitrary motion, can always be parametrized to inertial motion, where it does not radiate. The effects normally ascribed to acceleration of the electron itself, such as the Larmor formula, can all be explained by the boundary conditions, not by the source (the electron).
If the boundary condition, for example, includes absorption, that energy cannot return even if the electron is treated using quantum mechanics (QM) (without QM, a free electron cannot absorb a photon; in relativistic physics it is not possible to satisfy the energy and momentum conservation at the same time).