(What sort of direction is he taking the company) - my immediate thought would be to look after you to the extend you would be better off than going through the insurance.
If you are not going to be slighted by the MOT omission, personally I'd have all the worked quoted by a Toyota Garage (so long as its not too much that's going to write the car off and you cant buy it back - happened to me recently with my black 140) and go through the insurance- and if your body pain is due as a result of the accident then....
I wish you best of luck getting this resolved.
I'd have expected they would look after you for A: not wanting the bad publicity B: not wanting a claim in the "claims history" We all know how hire cars and company cars are driven, they are not in the same category as the care and attention our fold take in our cars, that makes it more painful when your pride and joy is damaged through no fault of your own. (still park mine at the back of every car park in the widest space.
I had an accident in a Golf GTI - many years back - after the accident I noticed the MOT had run out by 3 days. I was extremely worried to say the least - Nothing ever came of it and the insurers paid out.
You are doing the right thing by drilling down to a granular level, we all know the reputation of insurers that if they can wriggle out of paying out or paying out a reduced amount because of clause "X" they will. Sometimes it's best to air on the side of caution.
Do you know who their insurers are and their policy? ( if this has been mentioned, apologies)