As a former hiring manager, I can say I have never taken the time to google a candidate for lawsuits. If you settle out of court before going to trial, there is usually a nondisclosure and no public record.
1. Have your lawyer send a demand letter
2. Request a case management conference/arbitration
3. Reach settlement
4. File in court (filing now public)
5. Reach settlement (settlement private)
6. Court judgment (judgment public)
The goal is to stop at step 3 where it is not yet public. The company will not want the filing public because of bad publicity. If it gets this far, they would rather stop at step 5 than 6.