For an electron, how does QED maintain a constant rest mass and remove its classical radius from CEM? From classical electromagnetism, the classical radius of the electron is calculated to be
$$r_\text{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\text{e}} c^2} = 2.817 940 3227(19) \times 10^{-15} \text{ m}$$
Secondly, its rest mass is predicted to change when placed in a non-constant electric field as parts are accelerated differently to one another in its rest frame. QED on the other hand treats the electron as a point particle with a constant rest mass which so far have been increasingly confirmed experimentally. 
What additional physical processes, if any, come into play within QED to maintain the constant rest mass and point-like properties of the electron compared to within CEM?
 A: The issue is solved, in a way, by separating the "bare mass" that appears in the Lagrangian from the "physical mass" that we observe in experiments. The bare mass diverges under renormalization, irrespective of concerns about the energy content of the electric field around a point-like electron. A standard response to this is that the Lagrangian parameters aren't physically observable, anyway, they're just computational tools.
You can look at this problem from another point of view. individual electrons are excitations in a Fermionic field, usually one that obeys the Dirac equation in the appropriate limit. The electric field produced by a point-like excitation in that field has a divergent energy content in the associated electric field, but so what? That just means that point-like excitations aren't physically realizabile in that field and must, therefore, be computational tools, kind of like the virtual paths used when extremizing the action to get the equations of motion in classical physics.
