Any useful comments will be appreciated. Note that no new physics is required in this model, and all the laws of conservation (energy and momentum) are valid.

The ratio momentum over energy for a non relativistic electron is 2/v , where v is the velocity of the electron. This is orders of magnitude higher than 1/c, in the photon case. The chain reaction involving pair creation and annihilation, facilitated by the relativistic Doppler shift, where momentum is trasferred by the electron for each pair creation step, more than compensates for the energy required for the creation of the electron - positron pair. Your assumption that having the photon as the mediator of momentum transfer is the best you can do, is incorrect. Have you ever had a look at QFT mathematics? There are tomes of dense mathematics, just to understand the basics, and this is what we are talking about. Any attempt to dive into that on these posts would be a joke.

Answer for sanmanI don't know the answer to this one. We are still struggling with pair production (as you can see). I mentioned the Schwinger effect as an interesting venue of research related to all this, but as you can see from these discussions, we cannot handle even much simpler events.