The MOT invalidating your insurance is a very grey area. My insurance company (Direct line) state that the car must be roadworthy (So not necessarily needing an MOT).
I once got pulled for no MOT, and was charged with driving with no insurance. I contacted Direct Line and they provided me with a letter clearly stating that I was insured at the time I was stopped.

The procurator Fiscal decided that the case was still worth trying, and took me to court over it. I produced the letter, and a copy of my policy. And I was asked, how the insurance company could know the vehicle was roadworth, without a current MOT. It went back and forward between lawyers with lawyer speak for a while, and eventually I was let off, but given a very strict warning, and told that on this occasion, I would be let off, but if the same happened again, then I would most certainly be prosecuted etc etc.