No. Don’t pay for the background check. If your worth the hire the company should pay. Although most companies are blind and stupid and won’t hire a person who made mistakes.
If I'm understanding the question correctly, Skip the Dishes is holding your account because of an existing bad driving record. They are strict about their drivers having a clean driving record and if you are given a ticket during your employment they will fire you. So if they are requesting a background check, it's most likely it's now a part of their screening process.