The Thomas precession of the electron as it orbits around the nucleus of an atom is explained in SRT as the result of instantaneous “Lorentz Boosts” (infinitesimal Lorentz transformations). Lorentz Boosts are required to keep the electron in its own inertial frame. Goldstein  explains the process in the following words:
Consider a particle moving in the laboratory system with a velocity v that is not constant. Since the system in which the particle is at rest is accelerated with respect to the laboratory, the two systems should not be connected by a Lorentz transformation. We can circumvent this difficulty by a frequently used strategem (elevated by some to the status of an additional postulate of relativity). We imagine an infinity of inertial systems moving uniformly relative to the laboratory system, one of which instantaneously matches the velocity of the particle. The particle is thus instantaneously at rest in an inertial system that can be connected to the laboratory system by a Lorentz transformation. It is assumed that this Lorentz transformation will also describe the properties of the particle and its true rest system as seen from the laboratory system.
In the SRT model the precession results from the fact that successive Lorentz boosts are not collinear and the result is a rotation of the local reference frame. Magically, this rotation of the local reference frame results in a torque-free precession of an orbiting electron in the laboratory frame.
Instead of magic, the MLET explanation is that a real torque is generated when the force is non-gravitational and when the orbiting object is itself spinning (both apply in the case of an electron). When a spinning object is constrained by a non-gravitational force to follow a curved path, the spin velocity and the orbit velocity add as vectors. This means that the half of the spinning object where the speeds add will become contracted in length and will increase in inertial mass. Conversely, the half of the spinning object in which the spin velocity and orbital velocity combine to decrease the total speed will expand in length and decrease in inertial mass. These length and mass changes combine to cause a movement of the center of mass away from the center of the spinning object. When the force is non-gravitational and continues to act on the center of the object, a torque will be present which causes the Thomas precession.
The reasoning as to why the Thomas precession does not apply to objects orbiting under gravitational forces is a bit contrived for SRT. Specifically, it is argued that no gravitational force is present. Orbiting objects are assumed to be following a geodesic in space-time. That seems valid enough. But then why is the Lorentz transformation used to explain the aberration of starlight on the earth? These explanations seem to be mutually exclusive.
- Goldstein, Herbert (1980) Classical Mechanics, 2nd ed., Addison-Wesley, Reading, p 287.