The whole question is perfectly formulated, but allow me to reconsider these formulations nevertheless.
As far as I know, when electrons travel perpendicular to a uniform magnetic field, the Lorentz force makes the electron undergo circular motion.
The Lorentz force is the expression for the observed phenomenon that moving charges are deflected under the influence of a magnetic field (moving or not in relation to the charge, but necessarily not parallel to the trajectory of the charge). The Lorentz force is the result of the deflection of charges by magnetic fields and not the reason for the deflection.
Such an approach leads us to the question, how the magnetic field and a moving charge interact. With your next sentence you are close. You describe in detail what happens in the interaction.
As this electron undergoes circular motion, it emits EM radiation, so it goes through a spiral trajectory with a continuously decreasing orbital radius.
In addition to its electrical charge, the electron also has its own magnetic field. The electrons are magnetic dipoles. Now an interaction between the arriving electrons and the external magnetic field occurs naturally. The external field aligns these dipoles parallel to its own field.
The uncertainty in the description of what causes the Lorentz force (the deflection of the charges) is as follows.
Does the alignment of the magnetic dipole of the electron cause its deflection (gyroscope effect) and the lateral acceleration causes a photon to be emitted?
Or does the alignment of the magnetic dipole of the electron cause the emission of a photon and the electron is deflected by its recoil moment?
In any case, the relation between photon emission and loss of kinetic energy is obvious.
As this electron emits EM radiation, its energy should decrease.
It happens this way.
However, as far as I researched, the velocity of the electron should remain constant as the magnetic field does no work on the electron.
The magnetic field is like a catalyst or spring. It interacts again and again with the electrons magnetic dipoles. However, these electrons escape permanent alignment by photon emission.
Does this mean that the loss of the kinetic energy of the electron comes from its loss of mass?
If you apply the equivalence principle between energy and mass, then yes. But it would suffice to say that the loss of kinetic energy results from the emission and recoil of photons.