I would just second the advice to get a letter from your insurer in the U.S. If you had an agent this is much easier.
The U.S. doesn't have the same thing as this formally recognized "no claims bonus" in the UK. Don't worry about it. If you can get the letter to include the words to the effect of "no claims since" or a claims history, we found this to be most effective. Especially because I had to get two: one for car and one for motorcycle.
In the end, between the car and motorcycle, these letters have save us around a thousand pounds per year.