rainbow

One thing to remember is that everything is relative. So, although the thought experiment can place certain assumptions, they are relative to the observation of the OP.
The thought experiment presents a special presentation of a classic situation.
The thought experiment as placed HERE assumes too much- it states that the electron will not radiate at all from its own reference frame.
The very short answer is that from any reference point the electron will radiate. The difference is in how much can be measured depending on the change of momentum within each reference frame.
The measured radiation is proportional to the charge squared and inversely proportional to the mass squared.

IMO you would not expect an unaccelerated electron to radiate, even a little bit. It should just sit there. For one thing, it would have to get the energy from some where. Or am I missing something?