A running coupling constant breaks the classical limit and energy conservation?

UPDATE August 20, 2021: In the discussion below we did not analyze the fact that the fine structure constant α, too, changes if we change the charge e of the electron. The fine structure costant is approximately 1 / 137, and is proportional to e². If we increase e by a factor 12, then α will be larger than 1, and corrections will start to dominate quantum electrodynamics, making many formulas non-convergent. This requires a more thorough analysis.

The classical radius of the electron is 1 / 137 of the reduced Compton wavelength λ / (2 π) of the electron. Has Nature made α small enough, so that quantum mechanics hides the "inner field" of the electron closer than the classical radius?

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It is strange that people have not noticed the problems with a running coupling constant. If the force felt by a charge in a force field depends on the motion of the charge, that will break energy conservation. It also breaks the classical limit (Bohr's correspondence principle) of the quantum system, a fact about which we have already written in this blog.

https://www-thphys.physics.ox.ac.uk/people/FrancescoHautmann/Ralhep/index.html

Francesco Hautmann explains in the link the running of the coupling constant in QED.

       A ●  -------->

                                     impact paramer b

                      ● heavy charge

Let us have a charged particle A which passes another, very heavy charge. In QED, a running coupling constant means that the force felt by A depends on the amount of momentum it exchanges with the other charge. The coupling constant α is a function of the exchanged momentum q:

        α(q²).

If we make A go slower, then q² is larger, and A will move along a path which is different from what classical mechanics predicts. This breaks the classical limit.

Also, energy conservation is broken if the strength of a force felt by a charge depends on the motion of the charge. In QED, the coupling constant increases with |q|.

If we move the charge A slowly to a lower potential, there is a large exchange of momentum, and A will feel a strong force. We can then extract energy E. Then we move A quickly back to its original position. There is less exchange of momentum. The force is now weaker and we need to spend less energy E' < E. We have constructed a perpetuum mobile.

There are numerous problems with a running coupling constant. We have not seen any discussion of the problems in literature. As if researchers would have forgotten about conservation laws in physics.