I suspect the nuance and lesson from De Broglie's relationship, "lambda = h/p" which devolves classically to lambda (that is the rms positional uncertainty, or wavelength, equals Planck's constant divided by Newtonian momentum, where momentum p = mv). I suspect it is important to keep in mind that this relationship is "empirical", and is likely as fundamental as E = MC^{2}, and possibly even more so.

The Widom-Larsen theory thus considers, that the electrons are moving fast, so that they get heavy.

This is quite an orthogonal or even opposite implication from a relativistic mass increase with increased v. That is, strong confinement implies a proportionate mass increase. Essentially, slowing to a zero velocity implies an infinite average mass. Effective mass for electrons has many implications, and perhaps none apply here. But Widom-Larsen is a good place to start. That theory does imply effective mass can become useful mass in making up the mass - energy deficit normally realized as a neutrino in the reverse reaction.

I am certain is is not useful to devolve to Kepler and the Bohr atom, for many reasons well discussed in physics since the "ultraviolet catastrophe" beginning around 1900.