Yes.
Just an electron cannot emit Bremsstrahlung radiation. It needs to interact with other matter. And in this case, you are trading off kinetic energy to create the photon, not rest mass energy.
'My bad', I was not specific enough.
I wanted to point the rest mass of electron and proton do not change.
Then there is biding energy of hydrogen atom.
When electron emits a photon from hydrogen atom it is undergoing a 'Bremsstrahlung' trading off kinetic energy for photon emission in the hydrogen atom proton, electron system.
Yes. Even more importantly, though, it is part of the theory of special relativity. So if you want to evaluate SR, you need to account for it.
m is not constant. That's the whole point. You have started with a false assumption.
You are making an assumption that 'mass change', binding energy change is happening at the tip of the LED flashlight.
That is not the case, it is happening somewhere in batteries and these can be placed right in the center for the simplicity of argument.
If we do this then your argument about the mass change in relation to center of mass change is mute.
Edit: This is a thought experiment and we can have three LEDs arranged in a triangle at the tip to avoid any torquing when electrons do their job at the emission (if we have to go to such a detail
Edit 2: If we have to go there ...
If we are worried about EM field being transferred along wire from the batteries to the LEDs then we can have the same setup, creating symmetry, towards the opposite side.
Having said that, the LEDs at the back would not emit photons out but towards each other into a 'triangle' receiver to eliminate torquing at the back.
The back photons stay in the flashlight.
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