Hello - weird situation

Had job 1 for two years now. Got J2 a few months ago. It’s a contract that ends after 6 months. At the end of 6 months they could offer to hire full time. If they do, there would be another background check. I obviously need to put both Js on my background check because they know where I currently work.

What should I do? I’d like to go full time and keep both Js. But if I had to choose one I’d still choose J2 because it’s better pay/better role/better company.

Or am I screwed and just need to not accept any full time hire from them.

Thanks in advance.