The way I understand it, these people always do a thorough background check because they can't afford not to. So if you got as far as filling out paperwork or having the actual background check, they likely still have your social on file, not to mention your address and driver's license number. If you apply anew and any of the new info lines up with the old info, they will know who you are.
If you apply again, you will be asked about prior names as well. If they have all of this information on file and you leave off your old name on the new app, you can kiss Big Brothers good-bye. If they don't have it, they will still want to know about your old name, but you might be able to get creative about that, depending on the circumstances.
You can always call and ask how long they keep records on people who got as far as you did. They might want to look you up at that point, though, so be ready for that.
I had a situation kind of like this. I tried to donate blood when I was around twenty-five but was rejected. A quarter century later, I tried again because the prohibitions had lightened up some. The blood bank STILL had me on file, but as a female, and refused to change my gender marker. The SSN was a dead giveaway.
