But it's NOT good advertising. One car with some words on the side isn't going to drag in loads of work for them.
We get asked all the time about sponsoring cars, and everyone thinks that a few stickers and phone numbers equates to great advertising.
Their money would be far better spent on an ad in the local press. Your average bodyshop gets work from the surrounding area, so having a car that is at shows around the country means fuck all to them.
By all means send a letter, but you need to drastically improve what you're offering if you expect to get something for nothing.