You have thought all of these conundrums through much better than I have. But the math works fine.
"In Lorentz invariance, moving towards a light signal already in flight CANNOT hasten its arrival time, neither will moving away with the signal chasing after you delay its arrival time."
It is the inevitable ambiguity of language that complexifies complexity. Whenever you mention a problem that you have with MEE, you fail to specify which frame you are in. There is no change in interferometer arms for Michelson and Morley because they are in a rest frame. (There is a rotation effect, the Sagnac effect, but that has to do with the rotation of light, not its translation.)
Why you keep moving one arm or the other in your mind is how your mind works, but that is not how Lorentz invariance works.
Now when you introduce a moving frame, that does not change anything for the rest frame interferometer and Michelson and Morley can only know about the moving frame by looking outside of the rest frame. Once you introduce a moving frame and have that frame in communication with a rest frame, you now have the typical doppler shift effect, but with light pulses instead of continuous light waves.
It is actually easier to use light pulses for these kinds of discussions anyway.
Now in your mind, the moving observer simply by moving towards the source, has managed to get information from the source, a pulse, faster than the speed of light would allow. This is a very reasonable statement and has a basis in intuition, but the math does not work out. If you want to make your case, you must get information from the rest to the moving frame faster than c. Otherwise, everything that you have said is consistent with MEE.
Now in order to explain the limitation of c by MEE, you need to invoke the holy grail of time and space dilation and simultaneity and roll out your Minkowski diagrams and so on. Now the math monster rears its ugly head and many end up nodding off, since our language and common experience simply do not include the effects of MEE.
In fact, the space in between is contracting and so we are not receiving the pulse sooner because of time, but because of spatial contraction. Light pulses keep moving at c, but the space between rest and moving frames is now contracting at ~v2/c2, which we observe as a doppler shift, v/c. Oh, and that darn source clock has slowed down by the same factor.
So now we have a conflagration of spatial dilation, time dilation, and doppler shift and we are trying to describe this with a language and intuition that evolution has simply not provided beyond common predictions of action. Simultaneity down to better than a heartbeat is simply not needed for common experience.
So to avoid the ugly math, go to the geosynchronous orbit example that I gave before. Although there is the complexity of gravity effects, there is no motion doppler involved to complexify things. Synchronizing the moving and rest clocks compensates for gravity effects, but space and time dilation due to velocity, v2/c2, persist and are easy to calculate and communicate. Moreover, that is how our GPS system works and so we do not have to argue about an experiment done 150 years ago. The GPS performs this calculation many time a second for hundreds of millions of observers on the earth's surface.