I too can't believe you'd drive without insurance. All you have to do is ring your insurance company and tell them that from "X" time tomorrow, you'll be driving a different car and the Reg is "Z" and they will change it over automatically. If you'd never been insured before, you would have surely have gotten quotes and decided which company you were going to go with. In that scenario, you'd have phoned said insurance company and told them you were picking up a car tomorrow, quoting the Reg etc.
That said, I can't see how you should have any problem if the person said they were going to pay for it.
Recently, someone took all the paint off of the side of my car & they asked me would I go to a panel beater that was a friend of theirs to get the job fixed. I'd no problem with that, because after all that was all that I wanted. So, if they suggest they know someone that will do it cheaper than your estimate and for you to go there, I'd go. At least you are sure it will be done and that they payment will be arranged between the two of them.