You thought it was a warning? You have to be kidding me. I advise you not to tell any whoppers like that to your potential employer. Yes your suspended license conviction will show up on a background check. Will it keep you from getting a job? Only the car rental agency can tell you.
One's background is vague. There generally isn't one single database. In CA the DMV keeps a driving record (called an H6), the Department of Justice keeps records of all criminal convictions, as do all county courts. However, you weren't convicted of a crime, your license was suspended. You weren't charged or convicted of driving on a suspended license.
In a nutshell, you're fine. Just get the suspension cleared up as many employers need you to have a license. Also keep making payments on the ticket. Even if you did have a conviction for driving on a suspended license, it's hard to imagine that disqualifying you from a job.